Clustering Using REpresentatives

Both methods described above belong to a class of methods that is also called partitioning or optimization or partitioning-optimization techniques. They partition the set of objects into subsets according to some optimization criterion. Both methods use representative elements, in one case an object of the set to be clustered (the centrotype), in the other an object with real values for the variables that is not necessarily (and usually not) part of the objects to be clustered (the centroid). 

Figure 1 Binuclear iron core inserted over a) Z" and b) Z"0h clusters used to represent the 5MR portion of the ZSM-5 zeolite framework. T site numbers follow the ZSM-5 nomenclature. Pink balls are Al, grey balls are Si, blue balls are Fe, red balls are O and white balls are H.

Prediction of BOD value. In the ten clusters Identified by the K-means clustering procedure, two clusters were represented by chemicals with only low BOD values and one cluster with nearly all (18 of 19 or 95 %) high BOD values (Table III). Therefore, no discrimination was attempted within these clusters. In the remaining clusters there were 202 high BOD chemicals and 97 low BOD chemicals. Of these, approximately 75 % (152 of 202) were correctly classified Into the high BOD class, while 73 Z (71 of 97) were correctly classified Into the low BOD class. Using both the clustering and discrimination analyses, 77 % (170 of 220) and 78 % (93 of 120) of the chemicals In the data base were correctly classified. Within each of the clusters, between 2 and 4 molecular connectivity Indices were used In the final discriminant functions to separate the two classes of BOD. Within each cluster a different combination of variables were used as discriminators. Because of Che exploratory nature of this analysis, we lowered the F-ratlo Inclusion level Co 1.0. In several of the clusters, the F-ratlos for variables Included In Che discriminant functions were subsequently small(e.g., < 4.0). 

However the sample is prepared, we measure 13C spectra of one or more adsorbates on the catalyst, and then need to interpret the spectra to deduce the structure of adsorption complexes or reactive intermediates formed on the catalyst. In many cases the complexes and intermediates formed are unusual and exotic species for which the interpretation of the spectra may be far less than routine. This is where ab initio chemical shift calculations are essential. In diffraction methods, such as x-ray or neutron diffraction, one can more-or-less easily invert the experimental data to yield molecular structure. There is no straightforward relationship between chemical shift data and structure theoretical calculations provide the bridge between experiment and theory. In a typical study, we model the adsorbates on clusters that represent catalyst active sites, using experience and chemical intuition to create our initial structures. 

Figure 1. Embedded clusters used for simulating the anion vacancy (left) arid the tub divacancy (right). Oxygen arid magnesium ions are represented by white arid light grey circles respectively. The O ions at which the protori is bound are marked with Greek letters. The position of the missing oxygen is marked with V. The two Mg ions marked with an asterisk are included only when protori adsorption at O(fi) is considered.

Figure 8.2 Some model clusters used in the DV-Xa molecular orbital method. The effect of the structural optimization is taken into account. Gray atoms represent B, while the black and white ones are Si and H atoms, respectively. These arrangements are viewed along the [111] direction of the cluster.

Figure 1. Nickel clusters used in the present study. Gas phase clusters Niia Oh (a) and Ih (b) surface model clusters representing various adsorption sites on Ni(lOO) on-top position on Nig (c), fourfold-hollow position on Nig (d) and on Niir (e) and on Ni(lll) thr old-hollow position on Niio (f) and on-top position on Niio (f, upside down).

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