Model cyclic cluster

Figure 5.23 Model (HF) clusters (n = 3-5), with calculated net binding energies in parentheses. Only the cyclic pentamer (d) is a true equilibrium structure (see the text).

The application of this normalized, relative stress in Eq. (32) is essential for a constitutive formulation of cyclic cluster breakdown and re-aggregation during stress-strain cycles. It implies that the clusters are stretched in spatial directions with deu/dt>0, only, since AjII>0 holds due to the norm in Eq. (33). In the compression directions with ds /dt<0 re-aggregation of the filler particles takes place and the clusters are not deformed. An analytical model for the large strain non-linear behavior of the nominal stress oRjU(eu) of the rubber matrix will be considered in the next section. 

A series of papers by Beran et al. (102-112) was devoted to quantum-chemical studies of the cationic forms of zeolites using cluster models and the CNDO/2 technique. As a rule, cyclic clusters consisting of four or six T04 (T = Si, Al) tetrahedra were considered as models for four-membered (S, sites) and six-membered (S S, and S,r sites) oxygen windows in the zeolite structures. 

The difference in the electronic properties between the optimised linear and cyclic cluster models (eV)... 

In the theory of electronic structure two symmetric models of a boundless crystal are used or it is supposed that the crystal fills aU the space (model of an infinite crystal), or the fragment of a crystal of finite size (for example, in the form of a parallelepiped) with the identified opposite sides is considered. In the second case we say, that the crystal is modeled by a cyclic cluster which translations as a whole are equivalent to zero translation (Born-von Karman Periodic Boundary Conditions -PBC). Between these two models of a boundless crystal there exists a connection the infinite crystal can be considered as a limit of the sequence of cychc clusters with increasing volume. In a molecule, the number of electrons is fixed as the number of atoms is fixed. In the cyclic model of a crystal the number of atoms ( and thus the number of electrons) depends on the cyclic-cluster size and becomes infinite in the model of an infinite crystal. It makes changes, in comparison with molecules, to a one-electron density matrix of a crystal that now depends on the sizes of the cyclic cluster chosen (see Chap. 4). As a consequence, in calculations of the electronic structure of crystals it is necessary to investigate convergence of results with an increase of the cyclic cluster that models the crystal. For this purpose, the features of the symmetry of the crystal, connected with the presence of translations also are used. 

In (4.53), the index fi labels all AOs in the reference primitive unit cell (p = 1,2,..., M) and Rn is the translation vector of the direct lattice (for the reference primitive cell Rn = 0). The summation in (4.53) is snpposed to be made over the infinite direct lattice (in the model of the infinite crystal) or over the inner primitive translations ii of the cyclic cluster (in the cyclic model of a crystal). In the latter case, the sum of the two inner translation vectors fi -I- R = Ri may appear not to be the inner translation of the cychc cluster. However, the subtraction of the translation vector A of the cyclic cluster as a whole (in the cyclic model the vector A is... 

The above consideration can be interpreted as deduction of the cyclic cluster model of the infinite crystal when the Hartree-Fock LCAO method (or its semiempirical version with nonlocal exchange) is applied. 

The study of the approximate density matrix properties allowed the implementation of the cyclic cluster model in the Hartree- Fock LCAO calculations of crystalUne systems [100] based on the idempotency relations of the density matrix. The results... 

Hybrid approaches combining ab-initio or DFT and semiempirical approaches have become popular. As an example, we can refer to LEDO (hmited expansion of differential overlap) densities application to the density-functional theory of molecules [262]. This LEDO-DFT method should be well suited to the electronic-structure calculations of large molecules and in the anthors opinion its extension to Bloch states for periodic structures is straightforward. In the next sections we discuss the extension of CNDO and INDO methods to periodic stmctures - models of an infinite crystal and a cyclic cluster. 

Zero-differential overlap Approximation in Cyclic-cluster Model 211... 

The considered CNDO method for periodic systems formally corresponds to the model of an infinite crystal or its main region consisting of L primitive cells. This semiempirical scheme was also apphed for the cychc-cluster model of a crystal allowing the BZ summation to be removed from the two-electron part of matrix elements. In the next section we consider ZDO methods for the model of a cyclic cluster. 

Symmetry of Cyclic-cluster Model of Perfect Crystal... 

The idea to use relatively small cyclic clusters for comparative perfect-crystal and point-defect calculations appeared as an alternative to the molecular-cluster model in an attempt to handle explicitly the immediate environment of the chemisorbed atom on a crystalline surface [285] and the point defects in layered solids [286,287] or in a bulk crystal [288,289,292,293]. The cluster is formed by a manageable group of atoms around the defect and the difference between the molecular-cluster model (MCM) and the cychc cluster model (COM) is due to the choice of boundary conditions for the one-electron wavefunctions (MOs). Different notations of COM appeared in the literature molecular vmit ceU approach [288], small periodic cluster [286], large rmit cell [289,290]. We use here the cychc cluster notation. 

The CCM model allows real-space calculations (formaUy corresponding to the BZ center for the infinite crystal composed of the supercells). From this point of view the cyclic cluster was termed a quasimolecular large unit ceb [289] or a molecular unit ceb [288]. 

Semiempirical LCAO Methods in Cyclic-cluster Model... 

What happens when the cychc cluster is increased Depending on its shape and size different sets of fe-points are reproduced, but in the EHT matrix elements the number of interactions included (interaction radius) increases as the periodically reproduced atomic sites distance is defined by the translation vector of a cyclic cluster as a whole. It is important to reproduce in the cyclic-cluster calculations the states defining the bandgap. As the overlap matrix elements decay exponentially with the interatomic distance one obtains the convergence of results with increasing cyclic cluster. Of course, this convergence is slower the more diffuse are the AOs in the basis. Prom band-structure calculations it is known that for BNhex in the one-layer model the top of the valence band and the bottom of the conduction band are at the point P of the BZ reproduced in the cyclic cluster considered. 

Implementation of the Cyclic-cluster Model in MSINDO and Hartree-Fock LCAO Methods... 

For the small-size clusters (L = 4,8) the imbalance of the cychcity region and the interaction radius causes a huge difference between and IV as the sum of bond orders inside the interaction radius includes the neighboring cyclic clusters around the central cyclic cluster. The PBCs force atoms outside the central cyclic cluster to be equivalent to the corresponding atoms inside the cluster. At the same time the summation of bond orders inside a cyclicity region gives values that are largely model-size independent, as the DM is calculated for the cyclic cluster under consideration. 

Chemical Bonding in Cyclic-cluster Model Local Properties of Composite Crystalline Oxides... 

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