Cyclic „ clusters

When a defect is introduced into a crystalline environment, crystal translational symmetry can no longer be invoked to transform the problem into tractable form as is done in band-structure calculations. Most of the computational treatments of defects in semiconductors rely on approximations to the defect environment that fall into one of three categories cluster, supercell (or cyclic cluster), and Green s function. 

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. 

In fact, in the preferred cyclic geometry of Fig. 5.23(d) the orientation of adjacent dipoles is close (<20°) to the perpendicular angle at which dipole-dipole forces are vanishing. The strong preference for cyclic clusters is thus quite perplexing from a classical dipole-dipole viewpoint. 

An ab initio study of BF3 + (HF) clusters115 has shown that up to n 3, only weakly bonded van de Waals associations are found but with n = 4-7, cyclic clusters were formed in which BF3 is hydrogen bonded to HF with 3 fluorine atoms. Microwave studies116 and IR investigations117 show that the intermolecular BFbond is nevertheless somewhat shorter than expected on the basis of pure van de Waals interactions. 

In a recent thermodynamic and spectroscopic study Redington66 -68) evaluated AH and AS for the formation of cyclic and open-chain oligomers of hydrogen fluoride. His best fit to vapor density, heat capacity, excess entropy, excess enthalpy, and infra red absorption data shows a monotonous increase in AH per hydrogen bond with increasing number of HF molecules both for open-chain and cyclic clusters, using the relation... 

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 [P8W480i84] anion shows a cyclic cluster structure constructed by four P2W12O48 units derived from a Wells-Dawson anion by removal of six adjacent W06 octahedra. In the complex RE4(H20)28[KcP8W480i84(H4W40i2)2RE2(H20)io] (RE = La, Ce, Pr,... 

The size of the cyclic cluster under investigation was increased to six by White and Davidson . Because of their underlying interest in ice, the authors took the 0"0 distance as 2.75 A, rather than the larger separation in the gas phase r(OH) was fixed at 0.99 A. The structure examined was not of the classic type where all six molecules act as both donor and acceptor one molecule ( 5) serves as double donor and another ( 4) as double acceptor, as illustrated in Fig, 5.18. 

Tables 21 and 22 (and Figure 8) review Pt chemical shifts and metal-metal coupling constants of several chain clusters containing platinum, tungsten, nickel and molybdenum [119-121]. Likewise, Table 23 presents similar data for cyclic clusters containing eight metal atoms [120,122]. All these results are used to provide evidence of the structures, the symmetry and the isomerism of the clusters in solution.

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