Distance-averaging methods

Figure 7.5 Left, variation of the average plate height of fine-and coeurse-particle layers as a function of the solvent aigration distance and method of developaent. Right, relationship between the optiauB plate height and solvent migration distance for forced-flow development.

It is not possible to establish directly the value of the M-M distance corresponding to bond order of 1. Methods so easily applied to organic systems cannot be so readily applied here. First, the metallic radius for 12-coordinate metal is an average value, and second, as mentioned, the M-M distance (average) established for close-packed metals generally corresponds to a bond-order value of less than 1. At best only, the distance taken to correspond to a bond order of 1 is a crude approximation. Clearly, such arguments are enforced in any attempt to establish which correspond to bond orders of 2 or more. 

Distance-based methods require a definition of molecular similarity (or distance) in order to be able to select subsets of molecules that are maximally diverse with respect to each other or to select a subset that is representative of a larger chemical database. Ideally, to select a diverse subset of size k, all possible subsets of size k would be examined and a diversity measure of a subset (for example, average near neighbor similarity) could be used to select the most diverse subset. Unfortunately, this approach suffers from a combinatoric explosion in the number of subsets that must be examined and more computationally feasible approximations must be considered, a few of which are presented below. 

Compormd Central metal coordination number Nuclearity Average M-N distance (A) Method"... 

Table 9.3 A comparison of the single-linkage, complete-linkage and average-linkage cluster methods using the data in Table 92 The figures in parentheses indicate the distance between the clusters as they are formed In this particular case Ward s clustering follows the same order of cluster formation as the group average method.

Langer, K. (2001) A note on mean distances, 7 [mo6]. in substituted polyhedra, [(Mi, jM,t)06], in the crystal structures of oxygen based solid solutions local versus crystal averaging methods. Z Kristallogr., 216, 87-91. 

Let us consider the calculation of sensitivity threshold in the case when the cracks are revealing by PT method. Constant distance H between crack s walls along the whole defect s depth is assumed for the simplicity. The calculation procedure depends on the dispersity of dry developer s powder [1]. Simple formula has to be used in the case when developer s effective radius of pores IC, which depends mainly on average particle s size, is smaller than crack s width H. One can use formula (1) when Re is small enough being less than the value corresponding maximum sensitivity (0,25 - 1 pm). For example. Re = 0,25 pm in the case when fine-dispersed magnesia oxide powder is used as the developer. In this case minimum crack s width H that can be detected at prescribed depth lo is calculated as... 

Ihc complete neglect of differential overlap (CNDO) approach of Pople, Santry and Segal u as the first method to implement the zero-differential overlap approximation in a practical fashion [Pople et al. 1965]. To overcome the problems of rotational invariance, the two-clectron integrals (/c/c AA), where fi and A are on different atoms A and B, were set equal to. 1 parameter which depends only on the nature of the atoms A and B and the ii ilcniuclear distance, and not on the type of orbital. The parameter can be considered 1.0 be the average electrostatic repulsion between an electron on atom A and an electron on atom B. When both atomic orbitals are on the same atom the parameter is written , A tiiid represents the average electron-electron repulsion between two electrons on an aiom A. 

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