Quantum Atoms-in-Molecules Similarity

A different class of indices are those that behave like a distance. With these indices, a perfect similarity corresponds to a value 0, and no strict upper limit exists. The best-known of these D-class indices is the Euclidean distance introduced previously in Eqs. [7]-[13]. This Euclidean distance index has been extended to give an index for any operator used by chemists in evaluating the MQSMs by using 

This result yields the Euclidean distance when k = x = 2. 

Many more indices can be derived than those described here. Moreover, we can imagine many ways to obtain a D-class index by a mathematical manipulation of a C-class index and vice versa. Several examples of such transformations have been published by Carbo-Dorca et 

A field that has attracted special attention on different occasions is that of the similarity between two atoms located in two different molecules. Eor example, we can imagine calculating the similarity between two carbonyl carbon atoms, one in molecule A and one in molecule B, or calculating the similarity between a carbon atom in a molecule and the isolated atom, or between two different carbon atoms in the same molecule. From the perspective of molecular quantum similarity, calculating the similarity measure will require atomic electron densities within the molecule. Therefore, before turning to the similarity calculation, two methods for obtaining such densities will be briefly presented. 

The methods differ in a basic aspect. The first method, used by Boon et involves the Hirshfeld concept, previously described to some 

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