Kappa Shape Indices

The kappa shape indices [Hall and Kier 1991] are designed to characterise aspects of molecular shape by comparing a molecule with the "extreme shapes" that are possible for that number of atoms. As with the molecular connectivity indices, there are shape indices of various order (first, second, etc.). The first-order shape index involves a count over single bond fragments. The two extreme shapes are the linear molecule and the completely connected graph where every atom is connected to every other atom (Figure 3-3).

The first-order kappa index is defined as:

where 1 Pmax is the number of edges (or paths of length one) in the completely connected graph (see Figure 3-3); 1 Pmin is the number of bonds in the linear molecule; and 1P is the number of bonds in the molecule for which the shape index is being calculated. For a molecule containing A atoms,1 P^n equals (A — 1) and1 Pmax is A(A —1)/2, since all pairs of atoms are connected. Thus 1k becomes:

Figure 3-3. Extreme shapes used in the first- and second-order kappa indices for graphs containing four, five and six atoms. In both these cases the linear molecule corresponds to the minimum (middle column). The maximum for the first-order index corresponds to the completely connected graph (left-hand column) and for the second-order index to the star shape (right-hand column). (Redrawn from Leach 2001.)

Figure 3-3. Extreme shapes used in the first- and second-order kappa indices for graphs containing four, five and six atoms. In both these cases the linear molecule corresponds to the minimum (middle column). The maximum for the first-order index corresponds to the completely connected graph (left-hand column) and for the second-order index to the star shape (right-hand column). (Redrawn from Leach 2001.)

The second-order kappa index is determined by the count of two-bond paths, written 2 P. The maximum value corresponds to a "star" shape in which all atoms but one are adjacent to the central atom (2Pmax = (A — 1)(A — 2)/2, Figure 3-3) and the minimum value again corresponds to the linear molecule (2 Pmin = A — 2). The second-order shape index is then:

As with the molecular connectivity indices, higher-order shape indices have also been defined. The kappa indices themselves do not include any information about the identity of the atoms. This is achieved in the kappa-alpha indices where the alpha value for an atom i is a measure of its size relative to some standard (chosen to be the sp3-hybridised carbon):

rCsp3

Thus, the atom count A is adjusted for each non-sp3-hybridised carbon (either by an increment or a decrement). The kappa-alpha indices are calculated as follows:

where a is the sum of the ai's for all atoms in the molecule.

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