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 mu. The mu calculated by algebraic and vector semisums of the CT indices are defined. The model is generalized for molecules with heteroatoms and corrected for sp(3)-heteromolecules. The ability of the indices for the description of the molecular charge distribution is established by comparing them with mu of the valence-isoelectronic series of cyclopentadiene, benzene and styrene. Two CT indices, mu(vec) (vector semisum of vertex-pair mu) and mu(vec)V (valence mu(vec)) are proposed. The mu(vec) behaviour is intermediate between mu(vec) and mu(experiment). The correction is produced in the correct direction. The best results are obtained for the greatest group. Inclusion of the heteroatom in the pi-electron system is beneficial for the description of mu, owing to either the role of additional p and/or d orbitals provided by the heteroatom or the role of steric factors in the pi-electron conjugation. The steric effect is almost constant along the series and the dominating effect is electronic. Inclusion of the heteroatom enhances mu, 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].

The search for NLO organic materials with large values of the second hyperpolarizability (γ) attracted interest from the experimental and theoretical points of view [8,9]. Morley et al. calculated the first hyperpolaizability (β) of S-heteromolecules [10]. Zhao et al. computed γ for modified benzothiazoles, benzoxazoles and benzimidazoles [11]. Meyers et al. calculated the geometries and electronic and NLO properties of CT molecules based on the 2-methylene-2H-pyrrole repeating unit [12]. Li et al. computed structure–performance characteristics for β and γ of π-conjugated organic chromophores with the Pariser–Parr–Pople model [13]. Yeates et al. analyzed (X-ray) and calculated 2-(2-benzothiazolyl)-1-(2-thienyl)ethene as a model for high γ [14]. Gao et al. studied the effect of conjugated length on the computed β of organic molecules [15]. Tomonari et al. calculated the simplified sum-over-states and missing-orbital analysis on β and γ of benzene derivatives [16]. Glaser and Chen computed asymmetrization effects on structures of dipolar donor–acceptor-substituted molecular organic NLO materials [17]. Raptis et al. calculated the polarizability (α), β and γ of polysulfanes [18]. Rao and Bhanuprakash computed donor and acceptor organic molecules separated by a saturated C–C σ bond, which show large β [19]. Nakano et al. calculated γ of tetrathiapentalene and tetrathiafulvalene [20]. Cheng et al. computed α and β of H-silsesquioxanes [21]. Levitus et al. analyzed and calculated the photophysics of 1,4-bis(phenylethynyl)benzene [22] and 1,4-diethynyl- 2-fluorobenzene [23] to demarcate the effects of chromophore aggregation and planarization in poly(phenyleneethynylene)s. Öberg et al. computed β and γ of organic compounds [24].

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–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 sp3-heteromolecules. The next section introduces CT indices. Following that, the correction for sp3-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=l if i=j, “0” otherwise; l is the shortest edge count between vertices i and j [36]. In A, A=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-2. The intermediate matrix M is defined as the matrix product of A by D[-2]: The charge-transfer matrix C is defined as C = M – MT, where MT is the transpose of M [37]. By agreement, C=M. For i≠j, the C terms represent a measure of the intramolecular net charge transferred from atom j to i. The topological CT indices G are described as the sum, in absolute value, of the C terms defined for the vertices i,j placed at a topological distance D equal to k where N is the number of vertices in the graph, D 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 represents the sum of all the C terms, for every pair of vertices i and j at topological distance k. Other topological CT index, J, 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 μalg is defined as where C is the C 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 μalg is a graph invariant. An edge-to-edge analysis of μ suggests that each edge dipole moment μ connecting vertices i and j can be evaluated from the corresponding edge C index as

Each edge dipole can be associated with a vector μ in space. This vector has magnitude |μ|, 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 indexμvec is defined as the module of μ: and μvec 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 is redefined as to give the valence adjacency A where χ and χC are the electronegativities of heteroatom X and carbon, respectively, in Pauling units. Notice that the subtractive term keeps A=0 for the C atom (Equation 5). Moreover, the multiplicative factor reproduces A=2.2 for O, which was taken as standard. From the valence A, M and C matrices, μalg, μvec and topological CT indices G and J can be calculated by following the former procedure with the A matrix. The C, G, J, μalg and μvec descriptors are graph invariants. The main difference between μvec and μvec is that μvec is sensitive only to the steric effect of the heteroatoms, while μvec is sensitive to both electronic and steric effects.

Correction for sp3-Heteroatom-Containing Compounds

Kubinyi showed that the poor hydrogen-bond-formation capacity of the sp3-oxygen atoms that are directly linked to an sp2-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 sp3-X (–X–), X = O. Table 1 gives the molecular dipole moments μ for hydrocarbons and ethers calculated with different charge-transfer indices. The polarity decrease is also reflected by a significant decrease of the differential dipole moment (μether – μhydrocarbon) denoted as Δ(O – CH2). The Δ(O – CH2) μexperiment decreases with minus Δ(O – CH2) logP. The Δ(O – CH2) μvec does not show this diminution, while Δ(O – CH2) μvec gives very great values. However, Δ(O – CH2) μvec,corrected is of the same order of magnitude as both μexperiment and μMOPAC-AM1 references. As similar effects were shown for sp3-Si, P, Ge, As, Sn, Sb, Pb and Bi heteromolecules [34], Equation (6) is used for all sp3-X (–X–), X = O, Si, P, S, Ge, As, Se, Sn, Sb, Te, Pb, Bi, Po.

Table 1

Molecular dipole moment (in D) for hydrocarbons and ethers with charge indices.

Method	Compound	X = –CH2–	X = –O–	Δ(O – CH2)	Δ(O–CH2) log Pa
Vector semisum	Et–X–Et	0.407	0.436	0.029	–
	Phe–X–Et	0.739	0.659	-0.080	–
	Phe–X–Phe	0.427	0.333	-0.094	–
Valence vector semisum	Et–X–Et	0.407	2.854	2.447	–
	Phe–X–Et	0.739	2.621	1.882	–
	Phe–X–Phe	0.427	2.742	2.315	–
Corrected valence vectorsemisum	Et–X–Et	0.407	1.209	0.802	–
	Phe–X–Et	0.739	1.211	0.472	–
	Phe–X–Phe	0.427	1.204	0.777	–
Experimentb	Et–X–Et	0.087c	1.170	1.083	-2.50
	Phe–X–Et	0.350	1.410	1.060	-1.21
	Phe–X–Phe	0.260	1.150	0.890	0.07
AM1	Et–X–Et	0.006	1.246	1.240	–
	Phe–X–Et	0.257	1.264	1.007	–
	Phe–X–Phe	0.080	1.252	1.172	–

a P is the 1-octanol–water partition coefficient.

b Reference 40.

c Gaussian-2 composite ab initio method calculation taken from Reference 41.

Calculation Results and Discussion

The molecular CT indices G, J, G and J (with k<6) are listed in Table 2 for the valence-isoelectronic series of benzene (C6H6). As one might have expected, all the molecules show the same set of G (and, consequently, J) values. For instance, G1 is related to the degree of branching and G2 is related to the number of unsaturations in the molecule, which are constant throughout the series. On the other hand, G, which also depends on the electronegativity of the heteroatom through Equations (5–6), is influenced, in general, by the substitution.

In particular, G1 is related to the absolute differential electronegativity of the heteroatom |χ–χC| in the molecule. However, an exception occurs: G2V (and, as a result, J2) is equal throughout the series. C5H6Si, C5H6Ge, C5H6Sn and C5H6Pb show the same results for all G and J. This is due to the fact that Si, Ge, Sn and Pb have the same electonegativity (χSi=χGe=χSn=χPb=1.8). The same happens for C5H5Sb and C5H5Bi (χSb=χBi=1.9). Although in pyridine and C5H5As, the N and As atoms have the same absolute differential electronegativity |χN–χC|=|χAs–χC|=0.5, pyridine is calculated by Equation (5) while C5H5As, by Equation (6), and so G2V(pyridine) = 2G2V(C5H5As).

Charge indices up to the fifth order for the valence-isoelectronic series of benzene.a

Figure 1 illustrates the increment in the dipole moment of benzene in the presence of the heteroatom. The calculated μvec is of the same order of magnitude as μexperiment, while the calculated μvec=0 remains constant. Since μvec is sensitive to the steric effect but not to the electronic effect of the heteroatom, it is clear that the electronic effect (μvec) dominates over the steric one (μvec). In particular, the best results are obtained for the fourth long-period (Sn–Sb) and for the group-V heteromolecules.

Figure 1

Dipole moment of the valence-isolectronic series of benzene vs. the atomic number of the heteroatom. Experimental data from Reference 40. Points with Z = 14, 15, 32, 33, 50, 51, 82 and 83 are AM1 calculations.

Charge indices up to the fifth order for the valence-isoelectronic series of styrene.

The molecular CT indices G, J, G and G for the valence-isoelectronic series of styrene (C6H5–CH=CH2) are collected in Table 3. As expected, all the molecules show the same set of G and J values. However, G 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 (χS=χC=2.5). The same happens for the Si/Ge/Sn/Pb (χSi=χGe=χSn=χPb=1.8), P/Te (χP=χTe=2.1), As/Po (χAs=χPo=2.0) and Sb/Bi compounds (χSb=χBi=1.9).

Figure 2 shows the increase in the dipole moment of the valence-isoelectronic series of styrene when the heteroatom is present. Again, μexperiment and μvec vary in a similar fashion, while μvec remains almost constant (μvec~0.43D for the three groups IV–VI). The electronic effect of the heteroatom (μvec) dominates, in general, over the steric one (μvec). In particular, for thiobenzaldehyde (Z=16) the result of μvec = μvec (because χS=χC) 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.

Figure 2

Dipole moment of the valence-isolectronic series of styrene vs. the atomic number of the heteroatom. Point with Z = 6 from Reference 42; Z = 7, 14–16, 32–34, 51 and 52 are computed with AM1; Z = 8 from Reference 43; Z = 50, 82 and 83 are PM3 calculations.

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. Once more, μexperiment and μvec vary in a similar fashion while μvec remains almost constant (μvec~0.45D). The electronic effect of the heteroatom (μvec) dominates over the steric one (μvec). In particular, for thiophene (Z=16) μvec = μvec (because χS=χC). However, the μvec relative error for thiophene (10%) is even smaller than for cyclopentadiene (12%).

Figure 3

Dipole moment of the valence-isolectronic series of cyclopentadiene vs. the atomic number of the heteroatom. Point with Z = 15 is AM1 calculation.

Conclusions

The following conclusions can be made from this study:

The behaviour of μvec is intermediate between μvec and μexperiment and so the correction introduced with respect to μvec is produced in the correct direction. The best results are obtained for the greatest group that can be studied.

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.

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.

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., cyclopentadienering < C4SiH6 ring < C4GeH6 ring < C4SnH6 ring < C4PbH6 ring and benzenering < C5SiH6 ring < C5GeH6 ring < C5SnH6 ring < C5PbH6 ring.

Inclusion of the heteroatom increases μ, which is smaller for the benzene than for the styrene series. On going from styrene to C6H5–CH=SnH2, μexperiment increases by a factor of 41. Although there is a minor steric effect (μvec increases by a factor of 1.6), the major effect is electronic (μvec augments by a factor of 12). From μvec to μvec 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, μexperiment increases by a factor of 4. Although there is an antagonistic steric effect (in fact μvec decreases), the major effect is electronic (μvec is trebled).

 

Molecules	N	G1	G2	J1	J2
all molecules	6	0.0000	5.3333	0.0000	1.0667

 

Molecules	G1V	G3V	J1V	J3V
benzene	0.0000	0.0000	0.0000	0.0000
pyridine	2.2000	0.1222	0.4400	0.0244
C5SiH6	1.5400	0.0856	0.3080	0.0171
C5PH5	0.8800	0.0489	0.1760	0.0098
C5GeH6	1.5400	0.0856	0.3080	0.0171
C5AsH5	1.1000	0.0611	0.2200	0.0122
C5SnH6	1.5400	0.0856	0.3080	0.0171
C5SbH5	1.3200	0.0733	0.2640	0.0147
C5PbH6	1.5400	0.0856	0.3080	0.0171
C5BiH5	1.3200	0.0733	0.2640	0.0147

a G, J (i > 2), G, J (i > 3) are zero for all the entries; G2 = 5.3333, J2 = 1.0667.

 

Molecule	N	G1	G2	G3	G4	G5
all molecules	8	1.0000	6.8889	0.4375	0.2133	0.0625

 

Molecules	J1	J2	J3	J4	J5
all molecules	0.1429	0.9841	0.0625	0.0305	0.0089

 

Molecules	G1V	G2V	G3V	G4V	G5V
styrene	1.0000	6.8889	0.4375	0.2133	0.0625
benzaldimine	1.6000	7.1639	0.5569	0.2708	0.0185
benzaldehyde	2.7000	7.4389	0.8014	0.4083	0.0255
C6H5–CH=SiH2	2.5400	6.5039	0.5297	0.3436	0.1241
C6H5–CH=PH	1.8800	6.6689	0.4375	0.2611	0.0977
thiobenzaldehyde	1.0000	6.8889	0.4375	0.2133	0.0625
C6H5–CH=GeH2	2.5400	6.5039	0.5297	0.3436	0.1241
C6H5–CH=AsH	2.1000	6.6139	0.4375	0.2886	0.1065
C6H5–CH=Se	1.2200	6.8339	0.4375	0.2133	0.0713
C6H5–CH=SnH2	2.5400	6.5039	0.5297	0.3436	0.1241
C6H5–CH=SbH	2.3200	6.5589	0.4808	0.3161	0.1153
C6H5–CH=Te	1.8800	6.6689	0.4375	0.2611	0.0977
C6H5–CH=PbH2	2.5400	6.5039	0.5297	0.3436	0.1241
C6H5–CH=BiH	2.3200	6.5589	0.4808	0.3161	0.1153
C6H5–CH=Po	2.1000	6.6139	0.4375	0.2886	0.1065

 

Molecules	J1V	J2V	J3V	J4V	J5V
styrene	0.1429	0.9841	0.0625	0.0305	0.0089
benzaldimine	0.2286	1.0234	0.0796	0.0387	0.0026
benzaldehyde	0.3857	1.0627	0.1145	0.0583	0.0036
C6H5–CH=SiH2	0.3629	0.9291	0.0757	0.0491	0.0177
C6H5–CH=PH	0.2686	0.9527	0.0625	0.0373	0.0140
thiobenzaldehyde	0.1429	0.9841	0.0625	0.0305	0.0089
C6H5–CH=GeH2	0.3629	0.9291	0.0757	0.0491	0.0177
C6H5–CH=AsH	0.3000	0.9448	0.0625	0.0412	0.0152
C6H5–CH=Se	0.1743	0.9763	0.0625	0.0305	0.0102
C6H5–CH=SnH2	0.3629	0.9291	0.0757	0.0491	0.0177
C6H5–CH=SbH	0.3314	0.9370	0.0687	0.0452	0.0165
C6H5–CH=Te	0.2686	0.9527	0.0625	0.0373	0.0140
C6H5–CH=PbH2	0.3629	0.9291	0.0757	0.0491	0.0177
C6H5–CH=BiH	0.3314	0.9370	0.0687	0.0452	0.0165
C6H5–CH=Po	0.3000	0.9448	0.0625	0.0412	0.0152