Cluster approximation

Master equation approach exact relations and cluster approximations ... 

For below one monolayer the deposit may consist of an inhomogeneous mixture of a bare substrate and clusters approximately one monolayer thick. The state with the lowest chemical potential is represented by a line through the origin and tangent to the E[N) curve near the first local minimum, at Ni Nq. The slope of this tangent defines a chemical potential Thus, as N is increased from 0 to the fraction of the sub-... 

The result indicating that the interface metal atoms in the supported clusters approximated as Rhe are charged, whereas those one layer removed from the support surface are not, leads to the question of the charge of the metal atoms in a flat metal raft. The issue has been addressed in experiments with y-Al203-supported clusters approximated well as Res, which were formed from H3Re3(CO)i2 [34]. The rhenium atoms were inferred on the basis of... 

The calculation of the core levels of LajCuO discussed above was performed within the cluster approximation wherein the dependence of the spectral shape on the oxygen band-width is missed out. In order to include the effect of the oxygen 2p derived band on the spectral shape, the core-level photoemission calculation has to be performed within the impurity model which includes the O 2p band-width (instead of a single O 2p level). For reasonable values of the hybridization strength, t, the range of values of A is such that the Cu appears close to the bottom of the... 

The previous two sections of this review deal with classical simulation methods. A description of the activation of adsorbates by acidic sites, together with any bond breaking or bond formation that may take place, is the realm of quantum mechanical (QM) simulations. These types of calculations are particularly well-suited to zeolite-adsorbate systems when the cluster approximation is used. The active acidic site in the zeolite is modeled by a molecular cluster, formed by cutting out a small portion of... 

An extension of the coupled-cluster approximation to the non-equilibrium classical systems [43-45] has allowed to study asymptotics of bimolecular reactions. It resulted in a rather unexpected conclusion that now the generally-accepted time dependence of the A+B —> 0 reaction for d = 3, n(t) oc f-3/4, is only the pre-asymptotic stage, with the true asymptotics n(t) oc f 1 Similar technique was used also for the study of diffusion-limited aggregation and structure formation processes [47],... 

Abstract. An embedded-cluster approximation is adopted for simulating the heterolytic dissociation of hydrogen at two intrinsic defects on the (001) surface of magnesium oxide the isolated anion vacancy, and the tub divacancy. The dissociation process is shown to be critically dependent on the structure of the electrostatic field at the surface both as concerns energetics and final configuration. 

The present work continues our theoretical investigations [1] of this TKHS family. In contrast to [1] the Ising model is employed here to examine the possible types of deuteron ordering in the low-temperature phase (below 100 K) as well as to study the thermodynamic behavior of these materials on semi-quantitative level in the limits of the Bethe cluster approximation. 

Instead of the very demanding CCSDT calculations one often performs CCSD (T) (note the parentheses), in which the contribution of triple excitations is represented in an approximate way (not refined iteratively) this could be called coupled cluster approximate (or perturbative) triples. The quadratic configuration method (QCI) is very similar to the CC method. The most accurate implementation of this in common use is QCISD(T) (quadratic Cl singles, doubles, triples, with triple excitations treated in an approximate, non-iterative way). The CC method, which is usually only moderately slower than QCI (Table 5.6), is apparently better [102]. CCSD(T) calculations are, generally speaking, the current benchmark for practical molecular calculations on molecules of up to moderate size. 

Two specific approximations were developed to solve the problems of surface chemistry the periodic approximation, where quantum-chemical methods employ a periodic structure of the calculated system, and the cluster approximation, where the model of solid phase of finite size is created as a cutoff from the system of solid phase (it produces unsaturated dangling bonds at the border of cluster). The cluster approximation has been widely used for studying the interactions of molecules with all types of solids and their surfaces [24]. This approach is powerful in calculations of systems with deviations from the ideal periodic structure like doping and defects. Clay minerals are typical systems having such properties. 

Acetic acid and ethyl acetate can dissociate on copper through cleavage of the C-X bond adjacent to the carbonyl group, where Xrepresents either hydroxyl or alkoxy. The chemical behavior of supported copper was simulated by using a 13-atom copper cluster resembling an icosahedron (75). All the surface atoms in this cluster have the same coordination number of six (Fig. 16). Quantum chemical calculations within the cluster approximation depend on the number of surface and bulk atoms incorporated in the model therefore, results of these calculations should be used only to deduce trends in bonding and reactivity for various species adsorbed on copper. 

Stanton JF, Gauss J (1995) Perturbative treatment of the similarity transformed Hamiltonian in equation-of-motion coupled-cluster approximations. J Chem Phys 103 1064-1076. 

Certainly, the cluster calculations require some modification of quantum-chemical computational schemes. This is primarily connected with the main drawback of the cluster approximation, consisting of artificial scission of the chemical bonds between the cluster and the rest of the lattice. In most cases, however, such modifications do not involve the principal points, and therefore widely used quantum-chemical computer programs can serve as the basis for the cluster calculations. Some quantum-chemical methods used in... 

The calculation of x-ray emission spectra of molecules or solids are one of the most successful applications of the discrete variational (DV) Hartree-Fock-Slater (Xa) MO method using cluster approximation [8-10], which was originally coded by Ellis and his CO workers [11-14] based on Slater s Xa exchange potential [15]. The DV-Xa method has several advantages for the calculation of x-ray transition process as follows. 

The third advantage of the DV-Xa method is in the cluster approximation. The core hole state as the initial state of x-ray emission or final state of x-ray absorption is treated as an impurity atom in solid when using the band theory. However, since the core hole potential usually affects the next nearest neighbor atoms at most, it is enough for the calculation of x-ray emission spectra to use the clusters which include the second nearest neighbor atoms to the x-ray emitting atom as is described below. 

Cluster size for the calculation of infinite solids CmO The cluster size, when we calculate an infinite solid, is an important parameter using the cluster approximation. The size of the cluster or the termination method of the cluster (the charge of a cluster) has the largest effect on the accuracy of the DV-Xa calculation. 

An extended two-particle cluster approximation involving nearest-and next-nearest neighbor exchange for para- and ferromagnets (with particular reference to EuO, EuS, EuSe, and EuTe) has been developed (22). Heat capacity curves. Curie temperatures, magnetization curves. 

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