Valence Topological Charge-Transfer Indices for Reflecting Polarity: Correction for Heteromolecules

Valence topological charge-transfer (CT) indices are applied to the calculation of dipole moments μ. The μ calculated by algebraic and vector semisums of the CT indices are defined. The model is generalized for molecules with heteroatoms and corrected for sp3-heteromolecules. The ability of the indices for the description of the molecular charge distribution is established by comparing them with μ of the valence-isoelectronic series of cyclopentadiene, benzene and styrene. Two CT indices, μvec (vector semisum of vertex-pair μ) and μvecV (valence μvec) are proposed. The μvecV behaviour is intermediate between μvec and μexperiment. The correction is produced in the correct direction. The best results are obtained for the greatest group. Inclusion of the heteroatom in the π-electron system is beneficial for the description of μ, owing to either the role of additional p and/or d orbitals provided by the heteroatom or the role of steric factors in the π-electron conjugation. The steric effect is almost constant along the series and the dominating effect is electronic. Inclusion of the heteroatom enhances μ, which can improve the solubility of the molecule. For heteroatoms in the same group, the ring size and the degree of ring flattering are inversely proportional to their electronegativity.


Introduction
Homo and heterocycles were studied as models of fluorescers, organic conducting polymers or nonlinear optical (NLO) materials. New fluorescers contain heteroaromatic components. Some heterocycles recur often in industrial fluorescers. They do not fluoresce themselves, but have a fluorescence enhancing effect when coupled to conjugated systems. Reiser et al. measured the absorption and emission spectra, and fluorescence yields of aromatic benzoxazole derivatives [1]. Lippert et al. reviewed the photophysics of internal twisting [2]. Dey and Dogra measured and calculated the solvatochromism and prototropism in 2-(aminophenyl)benzothiazoles [3]. Catalán et al. studied the role of the torsion of the phenyl moiety in the mechanism of stimulated ultraviolet light generation in 2-phenylbenzazoles [4]. Levitus et al. performed photophysical measurements and semiempirical calculations with 1,4-bis(phenylethynyl)benzene [5]. Organic conducting polymers have a highly anisotropic quasi-one-dimensional (quasi-1D) structure similar to that of charge-transfer (CT) salts [6]. In the conducting state, both materials are ionic. In CT complexes conductivity is greater along the stacking direction, while in conducting polymers conductivity is higher along the chain direction. In these polymers, the chainlike structure leads to strong coupling of the electronic states to conformational excitations peculiar to 1D systems. The relatively weak interchain binding allows diffusion of dopant molecules into the structure, while the strong intrachain C-C bonds maintain the integrity of the polymer. The modulation of the electronic properties of conjugated polymers was studied through design of polymer backbone [7].
Organic electronic materials are conjugated solids where both optical absorption and charge transport are dominated by partly delocalized π and π * orbitals [25]. Organic photovoltaic materials differ from inorganic semiconductors in the following important respects. (1) Photogenerated excitations (excitons) are strongly bound and do not spontaneously dissociate into charge pairs. This means that carrier generation does not necessarily result for the absorption of light. (2) Charge transport proceeds by hopping between localized states, rather than transport within a band, and mobilities are low. (3) The spectral range of optical absorption is relatively narrow compared to the solar spectrum. (4) Absorption coefficients are high so that high optical densities can be achieved, at peak wavelength, with films less than 100nm thick. (5) Many materials are susceptible to degradation in the presence of oxygen or water. (6) As 1D semiconductors, their electronic and optical properties can be highly anisotropic. These properties impose some constraints on organic photovoltaic devices.
(1) A strong driving force such as an electric field should be present to break up the photogenerated excitons. (2) Low charge carrier mobilities limit the useful thickness of devices. (3) Limited light absorption across the solar spectrum limits the photocurrent. (4) Very thin devices mean interference effects can be important. (5) Photocurrent is sensitive to temperature through hopping transport.
In earlier publications, α of benzothiazole (A)-benzobisthiazole (B) A-B n -A (n≤13) oligomers was calculated and extrapolated to n→∞ [26]. Torsional effects were analyzed [27,28]. CT indices were brought to the calculation of the dipole moment µ of hydrocarbons [29]. The model was extended to heteroatoms [30]. An index inspired by plastic evolution improved the results [31,32]. The method was applied to the valence-isoelectronic series of benzene and styrene (2-4 molecules) [33,34]. This study presents a reparametrization of the method for sp 3 -heteromolecules. The next section introduces CT indices. Following that, the correction for sp 3 -heteromolecules is presented. Next, the results are discussed. The last section summarizes the conclusions.

Topological Charge-Transfer Indices
The most important matrices that delineate the labelled chemical graph are the adjacency (A) [35] and the distance (D) matrices, wherein D ij =l ij if i=j, "0" otherwise; l ij is the shortest edge count between vertices i and j [36]. In A, A ij =1 if vertices i and j are adjacent, "0" otherwise. The D [-2] matrix is the matrix whose elements are the squares of the reciprocal distances D ij The charge-transfer matrix C is defined as C = M -M T , where M T is the transpose of M [37]. By agreement, C ii =M ii . For i≠j, the C ij terms represent a measure of the intramolecular net charge transferred from atom j to i. The topological CT indices G k are described as the sum, in absolute value, of the C ij terms defined for the vertices i,j placed at a topological distance D ij equal to k where N is the number of vertices in the graph, D ij are the entries of the D matrix, and δ is the Kronecker δ function, being δ=1 for i=j and δ=0 for i≠j. G k represents the sum of all the C ij terms, for every pair of vertices i and j at topological distance k. Other topological CT index, J k , I defined as: This index represents the mean value of the CT for each edge, since the number of edges for acyclic compounds is N-1.
The algebraic semisum CT index µ alg is defined as where C e is the C ij index for vertices i and j connected by edge e [29]. The sum extends for all pairs of adjacent vertices in the molecular graph and µ alg is a graph invariant. An edge-to-edge analysis of µ suggests that each edge dipole moment µ e connecting vertices i and j can be evaluated from the corresponding edge C e index as Each edge dipole can be associated with a vector µ e in space. This vector has magnitude |µ e |, lies in the edge e connecting vertices i and j, and its direction is from j to i. The molecular dipole moment vector µ results the vector sum of the edge dipole moments as summed for all the m edges in the molecular graph. The vector semisum CT index µ vec is defined as the module of µ: and µ vec is a graph invariant.
When heteroatoms are present, some way of discriminating atoms of different kinds needs to be considered [38]. In valence CT indices terms, the presence of each heteroatom is taken into account by introducing its electronegativity value in the corresponding entry of the main diagonal of the adjacency matrix A. For each heteroatom X, its entry A ii is redefined as and µ vec V descriptors are graph invariants. The main difference between µ vec and µ vec V is that µ vec is sensitive only to the steric effect of the heteroatoms, while µ vec V is sensitive to both electronic and steric effects.

Correction for sp 3 -Heteroatom-Containing Compounds
Kubinyi showed that the poor hydrogen-bond-formation capacity of the sp 3 -oxygen atoms that are directly linked to an sp 2 -carbon atom (like in esters, aromatic ethers and furans) is also reflected by a significant decrease of their polarity (MedChem database 1-octanol-water partition coefficient, P) in going from aliphatic to araliphatic and to aromatic ethers R-O-R' (Table 1) [39]. Therefore, in this study it is suggested to halve the factor in Equation (5) as for sp 3 -X (-X-), X = O. Table 1 gives the molecular dipole moments µ for hydrocarbons and ethers shown for sp 3 -Si, P, Ge, As, Sn, Sb, Pb and Bi heteromolecules [34], Equation (6) is used for all sp 3 -X (-X-), X = O, Si, P, S, Ge, As, Se, Sn, Sb, Te, Pb, Bi, Po.

Calculation Results and Discussion
The molecular CT indices G k , J k , G k V and J k V (with k<6) are listed in Table 2 for the valenceisoelectronic series of benzene (C 6 H 6 ). As one might have expected, all the molecules show the same set of G k (and, consequently, J k ) values. For instance, G 1 is related to the degree of branching and G 2 is related to the number of unsaturations in the molecule, which are constant throughout the series. On the other hand, G k V , which also depends on the electronegativity of the heteroatom through Equations (5-6), is influenced, in general, by the substitution.
In particular, G 1 V is related to the absolute differential electronegativity of the heteroatom |χ X -χ C | in the molecule. However, an exception occurs: G 2 V (and, as a result, J 2 V ) is equal throughout the series. C 5 H 6 Si, C 5 H 6 Ge, C 5 H 6 Sn and C 5 H 6 Pb show the same results for all G k V and J k V . This is due to the fact that Si, Ge, Sn and Pb have the same electonegativity (χ Si =χ Ge =χ Sn =χ Pb =1.8). The same happens for C 5 H 5 Sb and C 5 H 5 Bi (χ Sb =χ Bi =1.9). Although in pyridine and C 5 H 5 As, the N and As atoms have the same absolute differential electronegativity |χ N -χ C |=⏐χ As -χ C ⏐=0.5, pyridine is calculated by Equation (5) while C 5 H 5 As, by Equation (6), and so G 2 V (pyridine) = 2G 2 V (C 5 H 5 As).
are zero for all the entries; G 2 V = 5.3333, J 2 V = 1.0667. Figure 1 illustrates the increment in the dipole moment of benzene in the presence of the heteroatom. The calculated µ vec V is of the same order of magnitude as µ experiment , while the calculated µ vec =0 remains constant. Since µ vec is sensitive to the steric effect but not to the electronic effect of the heteroatom, it is clear that the electronic effect (µ vec V ) dominates over the steric one (µ vec ). In particular, the best results are obtained for the fourth long-period (Sn-Sb) and for the group-V heteromolecules.  The molecular CT indices G k , J k , G k V and G k V for the valence-isoelectronic series of styrene (C 6 H 5 -CH=CH 2 ) are collected in Table 3. As expected, all the molecules show the same set of G k and J k values. However, G k V are influenced by the atomic number of the heteroatom. In particular, the results for thiobenzaldehyde are equal to those for styrene. This is because the electronegativity for the S atom has been taken equal to that of C (χ S =χ C =2.5). The same happens for the Si/Ge/Sn/Pb (χ Si =χ Ge =χ Sn =χ Pb =1.8), P/Te (χ P =χ Te =2.1), As/Po (χ As =χ Po =2.0) and Sb/Bi compounds (χ Sb =χ Bi =1.9). Figure 2 shows the increase in the dipole moment of the valence-isoelectronic series of styrene when the heteroatom is present. Again, µ experiment and µ vec V vary in a similar fashion, while µ vec remains almost constant (µ vec~0 .43D for the three groups IV-VI). The electronic effect of the heteroatom (µ vec V ) dominates, in general, over the steric one (µ vec ). In particular, for thiobenzaldehyde (Z=16) the result of µ vec V = µ vec (because χ S =χ C ) should be taken with care. It is an artefact of the model for S-heteromolecules. Furthermore, the best results are obtained, in general, for the fourth long-period (Sn-Te) and for the group-VI heteromolecules. In order to test the model for other S-containing heterocycle, Figure 3 displays the increase in the dipole moment of the valence-isoelectronic series of cyclopentadiene when the heteroatom is present.

Conclusions
The following conclusions can be made from this study: 1. The behaviour of µ vec V is intermediate between µ vec and µ experiment and so the correction introduced with respect to µ vec is produced in the correct direction. The best results are obtained for the greatest group that can be studied.
2. Inclusion of the heteroatom in the π-electron system is beneficial for the description of the dipole moment, owing to either the role of additional p and/or d orbitals provided by the heteroatom or the role of steric factors in the π-electron conjugation. The analysis of both electronic and steric factors in µ caused by the presence of the heteroatom shows that the electronic factor dominates over the steric one. Work is in progress on the calculation of the dipole moments of a homologous series of 4-alkylanilines, which are percutaneous enhancers of transdermal-delivery drugs.
3. Inclusion of the heteroatom enhances µ with the only exception of the insertion of the Si atom in styrene. In turn, the increase in µ can improve the solubility of the molecule.
4. For heteroatoms in the same group of the periodic table, the ring size and the degree of ring flattering are inversely proportional to the electronegativity of the hetetoatom, e.g., cyclopentadiene ring < C 4 SiH 6 ring < C 4 GeH 6 ring < C 4 SnH 6 ring < C 4 PbH 6 ring and benzene ring < C 5 SiH 6 ring < C 5 GeH 6 ring < C 5 SnH 6 ring < C 5 PbH 6 ring .
5. Inclusion of the heteroatom increases µ, which is smaller for the benzene than for the styrene series. On going from styrene to C 6 H 5 -CH=SnH 2 , µ experiment increases by a factor of 41.
Although there is a minor steric effect (µ vec increases by a factor of 1.6), the major effect is electronic (µ vec V augments by a factor of 12). From µ vec to µ vec V the introduced correction is produced in the correct direction. However, the result for thiobenzaldehyde is uncertain. Work is in progress with the correct parameterization of the method for the S atom. On going from cyclopentadiene to pyrrole, µ experiment increases by a factor of 4. Although there is an antagonistic steric effect (in fact µ vec decreases), the major effect is electronic (µ vec V is trebled).