The E-state Index

Kier and Hall noticed that the quantity (S -S) jn, where n is the principal quantum number and 5 is computed with Eq. (2), correlates with the Mulliken-Jaffe electronegativities [19, 20]. This correlation suggested an application of the valence delta index to the computation of the electronic state of an atom. The index (5 -5)/n defines the Kier-Hall electronegativity KHE and it is used also to define the hydrogen E-state (HE-state) index. 

The E-state indices are atomic descriptors composed of an intrinsic state value I and a perturbation AI that measures the interactions with all other atoms in a molecule. The Kier-Hall electronegativity is the starting point in the definition of the intrinsic state of an atom, which encodes its potential for electronic interactions and its connectivity with adjacent atoms. The intrinsic state of an atom i is [19, 21]  

The second contribution to the E-state index comes from the interactions between an atom i and all other atoms in the molecular graph. The perturbation on the intrinsic state value / of atom i due to another atomj depends on the difference between the corresponding intrinsic state values, (f- /,), and on the graph distance between atoms i andj. The overall perturbation on the intrinsic state value I of atom i is  

Finally, the E-state index S of atom i is the sum of the intrinsic state and of the perturbation term  

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The electrotopological state (E-state) indices are a group of atom level descriptors that are calculated for each atom (such as >C<, >N-, =0) or hydride group (such as -CH3, >NH, UOH) in the molecule. For simplicity, these groups are all termed atoms (Rose et al., 2002). The E-state index for an atom i in a molecule, S(i), is composed of an intrinsic state, I, plus the sum of perturbations, IM. from all other atoms in the molecule ... 

From the definition of intrinsic states and field effects, it can be seen that large positive values of E-states S relate to atoms of high electronegativity and/or terminal atoms or atoms that lie on the mantle of the molecule small or negative E-state values correspond to atoms possessing only o electrons and/or buried in the interior of the molecule or close to higher electronegative atoms. Therefore, the E-state index is a measure of the electronic accessibility of an atom and can be interpreted as a probability of interaetion with another molecule. However, the index cannot be considered a pure electronic descriptor it is, in fact, a descriptor of atom polarity and steric accessibility. 

The 5 connectivity index (atom degree), that has a central role in computing the E-state, was used in the definihon of the Zagreb topological indices [11]. Randle modified the Zagreb index M2 to obtain the connectivity index % [12]. 

Voelkel used the formula of the J index [24] to define the E-state topological... 

The E-state indices may define chemical spaces that are relevant in similarity/ diversity search in chemical databases. This similarity search is based on atom-type E-state indices computed for the query molecule [55]. Each E-state index is converted to a z score, Z =(% -p )/0 , where is the ith E-state atomic index, p is its mean and O is its standard deviation in the entire database. The similarity was computed with the EucHdean distance and with the cosine index and the database used was the Pomona MedChem database, which contains 21000 chemicals. Tests performed for the antiinflamatory drug prednisone and the antimalarial dmg mefloquine as query molecules demonstrated that the chemicals space defined by E-state indices is efficient in identifying similar compounds from drug and drug-tike databases. 

The data containing 324 descriptor values of 88 molecules was given as an input to VSMP program, to build models based on three and four descriptors, keeping the interdescriptor correlation below 0.75. The best three-descriptors model, Eq. 80, was based on descriptors 254 (atomic type E-state index), 311 (AlogP98), and 320 (2D Van der Waals surface area) with a correlation coefficient, r, of 0.8425, and the cross-validated correlation coefficient, q, of 0.8239. The correlation coefficients of the other two VSMP models, Eqs. 81 and 82 were 0.8411 and 0.8329, respectively. Significantly, the descriptors 254 and 311 were selected in all the best three-descriptors models of VSMP. The three descriptors, in the models 80, 81, and 82 were 320, 144 (Kappa shape index of order 1), and 30 (topological Xu index), respectively. 

These summarize topological information about a molecule with atomic properties [39]. A molecular E-state index is expressed as a sum of atomic E-state indices, which are composed of two parts. First is the intrinsic atomic part, and second is perturbation, which depends on its neighborhood (other atoms in the structure). The intrinsic part includes information about the a- and 7r-orbitals, lone pairs, hydrogen atoms attached to heavy atoms, and the principal quantum numbers of valence electrons. The perturbation part is a sum of all other atomic parts modified by fimction, which descends with distance. 

Based on the same approach used to define -state indices, a bond E-state index BSh was also tentatively proposed as ... 

Methods Using 2D and ID Descriptors A good number of articles on aqueous solubility used a nonlinear method of data analysis, in particular, for methods developed with ID and 2D descriptors. Huuskonen [16] used E-state indexes [52,53] and several other topological indexes, with a total of 30 indexes, to develop his models. The predicted results for the 413 test set, SE = 0.71, calculated with MLRA were improved with a neural network, resulting in SE = 0.6. Tetko [17] noticed that E-state indexes represent a complete system of descriptors for molecules, and thus only these descriptors are sufficient to develop the aqueous solubility model. Indeed the model developed by the authors using exclusively E-state indexes provides similar results when compared to the model of Huuskonen [16]. Later on, the model was redeveloped using the Associative Neural Network (ASNN) method [54],... 

E-state indexes were used by Votano et al. [55] to develop two models (one for aromatic and one for nonaromatic compounds) for a dataset of 5694 molecules. A comparison of PLS, MLRA, and ANN models for their prediction of the test set molecules clearly indicated the advantage of the nonlinear methods. The neural networks calculated MAE = 0.62 and MAE = 0.56 for aromatic and nonaromatic sets, respectively. The second-best results for the same test sets, MAE = 0.76 and MAE = 0.66, were calculated using the MLRA model. 

The topological state index (T ) of an atom represents the position of the atom in the scaffold of the molecular structure in relation to aU other atoms of the molecule (hut based on topology, i.e., on the connectivity and not on the three-dimensional structure) [54], Chemically and topologically equivalent atoms have identical indices. 

The dectrotopological state index (Si) is an extension of the purely topological index [55]. Electronic properties (i.e., the charge distribution in the molecule) are also considered. Atoms with identical T may differ in their Si. Electrotopological state indices have been used successfully to predict physico-chemical data, such as basicity (pJCa) or Hpophilicity (logP), as well as for quantitative structure-activity relationship (QSAR) or QSPR studies. 

The E-state topological parameter, denoted as TI , is derived by applying the Ivanciuc-Balaban operator to the —> Estate index values used to characterize molecule atoms [Voelkel, 1994] ... 

The definition of these 30 indices is shown in Tables 7.1 and 7.2, where the A index encodes the number of electronegative atoms ( ea), while the E index encodes the sum of the S-State index for the electronegative atoms, N, 0, F, Cl, Br is the average value for a specific type of atom, sulphur has not been considered as an electronegative atom). The rescaling procedure brings about... 

The full combinatorial technique finds also an improved semiempirical combinations with five indices (see later on), which include Tf, and T im but we prefer the more economical three-index combination. The improvement relatively to the greedy algorithm is mainly in The strongest correlations are r(Ti, = 0.60, the strongest correlation of the dielectric constant is, r e, T /m) = 0.93. Notice that the T /m tettn encodes the information about the molar mass but also about the overall -State index of the electronegative atoms. It reflects the charge distribution due to these atoms, normalized to the molar mass, a no trivial choice. 

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