Kappa indices

The second-order kappa index is determined by the count of two-bond paths, written P. The maximum value is expressed by a star shape = (A — 1) (A — 2) fl) and the mini-... 

To illustrate how different m(X ) and x may happen to be, let s consider as a specific example (others can be found in Saraiva and Stephanopoulos, 1992c) a Kraft pulp digester. The performance metric y, that one wishes to minimize, is determined by the kappa index of the pulp produced and the cooking yield. Two decision variables are considered H-factor (xj), and alkali charge (X2). Furthermore, we will assume as perfect an available deterministic empirical model (Saraiva and Stephanopoulos, 1992c), /, which expresses y as function of x, i.e., that y =/(xi, X2) is perfectly known. 

Kier, L.B. (1987b). Inclusion of Symmetry as a Shape Attribute in Kappa Index Analysis. Quant Struct.-Act.Relat, 6, 8- 2. 

Example 2 illustrates how coordination by the two terminal primary amino groups of the ligand is indicated by placing the kappa index after the substituent group name and within the effect of the bis doubling prefix. The appearance of the simple index kN after the ethane-1,2-diamine indicates the binding by only one of the two equivalent secondary amino nitrogen atoms. 

Tridentate chelation by the tetrafunctional macrocycle in Example 4 is shown by the kappa index following the ligand name. The ligand locants are required in order to distinguish this complex from those where the central atom is bound to other combinations of the four potential donor atoms. 

The mixed sulfur-oxygen cyclic polyether l,7,13-trioxa-4,10,16-trithiacyclooctadecane might chelate to alkali metals only through its oxygen atoms and to second-row transition elements only through its sulfur atoms. The corresponding kappa indexes for such chelate complexes would be k]O1,07,013 and K3 1, 10, 16, respectively. 

Examples 9-11 illustrate three modes of chelation of the ligand /V- W-(2-aminoethyl)-/V. -S -diphenylsulfonodiimidoyljbenzenimidamide. The use of kappa indexes allows these binding modes (and others) to be distinguished and identified, in spite of the abundance of heteroatoms that could coordinate. 

These topological indexes, based on the molecular connectivity approach, include three types the ""Xr molecular connectivity chi indexes that characterize the structural attributes of molecules, the ""k kappa indexes of molecular shape, and the topological equivalence state T values that individually characterize atoms and groups in the molecular skeleton and are used primarily to determine chemically equivalent atoms within a molecule. A further development of this approach has led to the electrotopological state atom indexes, which will not be discussed here but will be presented elsewhere. Molecular connectivity chi indexes are discussed in the first part of this paper along with illustrative applications. Then kappa shape indexes are discussed. The topological state index is discussed in the final section. 

Graph-based structure analyses had not been used for the specific objective of encoding relative shape. There was the frequent inference that indexes derived from these methods encode shape information, although none had been conceived with this specific objective in mind. It is from the graph of a molecule that we have derived the kappa indexes with the objective of encoding relative shape. 

Table 3 Selected Graphs to Illustrate the Range of Kappa Index Values...

One application of kappa indexes is quantitation of shape similarity in contrast to the absolute quantitation of size. The kappa values permit a rational prediction of which molecules have a high degree of shape similarity. Thus, n-pentane, diethylether, and diethylamine have similar kappa values. Electronic structure is information to be encoded into other indexes. The kappa values predict that these three molecules have very similar shape. 

Each of the structural attributes in the definition above has been incorporated into our analysis of shape using the kappa indexes. In particular, the count of atoms and the relative cyclicity of molecules have been shown to be... 

A correlation was found between the Taft , values and three kappa indexes ... 

The kappa shape indexes are relatively new descriptors, and few studies have been reported to date. It is expected that the kappa indexes will generally be used along with other topological indexes in QSAR equations rather than by themselves as in most of the examples above. Shape is not the sole determinant... 

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