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

## Abstract

**:**

^{3}-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, μ

_{vec}(vector semisum of vertex-pair μ) and μ

_{vec}

^{V}(valence μ

_{vec}) are proposed. The μ

_{vec}

^{V}behaviour is intermediate between μ

_{vec}and μ

_{experiment}. 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

^{*}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.

_{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

**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}

^{-2}. The intermediate matrix

**M**is defined as the matrix product of

**A**by

**D**

^{[-2]}:

**M**=

**AD**

^{[-2]}

**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

_{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:

_{alg}is defined as

^{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

**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

_{vec}is defined as the module of

**μ**:

_{vec}is a graph invariant.

**A**. For each heteroatom X, its entry A

_{ii}is redefined as

**A**

^{V}matrix where χ

_{X}and χ

_{C}are the electronegativities of heteroatom X and carbon, respectively, in Pauling units. Notice that the subtractive term keeps A

_{ii}

^{V}=0 for the C atom (Equation 5). Moreover, the multiplicative factor reproduces A

_{ii}

^{V}=2.2 for O, which was taken as standard. From the valence

**A**

^{V},

**M**

^{V}and

**C**

^{V}matrices, μ

_{alg}

^{V}, μ

_{vec}

^{V}and topological CT indices G

_{k}

^{V}and J

_{k}

^{V}can be calculated by following the former procedure with the

**A**

^{V}matrix. The C

_{ii}

^{V}, G

_{k}

^{V}, J

_{k}

^{V}, μ

_{alg}

^{V}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

^{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

^{3}-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 (μ

_{ether}– μ

_{hydrocarbon}) denoted as Δ(O – CH

_{2}). The Δ(O – CH

_{2}) μ

_{experiment}decreases with minus Δ(O – CH

_{2}) logP. The Δ(O – CH

_{2}) μ

_{vec}does not show this diminution, while Δ(O – CH

_{2}) μ

_{vec}

^{V}gives very great values. However, Δ(O – CH

_{2}) μ

_{vec}

^{V}

^{,corrected}is of the same order of magnitude as both μ

_{experiment}and μ

_{MOPAC-AM1}references. As similar effects were 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.

Method | Compound | X = –CH_{2}– | X = –O– | Δ(O – CH_{2}) | Δ(O–CH_{2}) log P^{a} |
---|---|---|---|---|---|

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 | – | |

Experiment^{b} | Et–X–Et | 0.087^{c} | 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

_{k}, J

_{k}, G

_{k}

^{V}and J

_{k}

^{V}(with k<6) are listed in Table 2 for the valence-isoelectronic 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.

_{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).

Molecules | N | G_{1} | G_{2} | J_{1} | J_{2} |
---|---|---|---|---|---|

all molecules | 6 | 0.0000 | 5.3333 | 0.0000 | 1.0667 |

Molecules | G_{1}^{V} | G_{3}^{V} | J_{1}^{V} | J_{3}^{V} |
---|---|---|---|---|

benzene | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

pyridine | 2.2000 | 0.1222 | 0.4400 | 0.0244 |

C_{5}SiH_{6} | 1.5400 | 0.0856 | 0.3080 | 0.0171 |

C_{5}PH_{5} | 0.8800 | 0.0489 | 0.1760 | 0.0098 |

C_{5}GeH_{6} | 1.5400 | 0.0856 | 0.3080 | 0.0171 |

C_{5}AsH_{5} | 1.1000 | 0.0611 | 0.2200 | 0.0122 |

C_{5}SnH_{6} | 1.5400 | 0.0856 | 0.3080 | 0.0171 |

C_{5}SbH_{5} | 1.3200 | 0.0733 | 0.2640 | 0.0147 |

C_{5}PbH_{6} | 1.5400 | 0.0856 | 0.3080 | 0.0171 |

C_{5}BiH_{5} | 1.3200 | 0.0733 | 0.2640 | 0.0147 |

^{a}G

_{i}, J

_{i}(i > 2), G

_{i}

^{V}, J

_{i}

^{V}(i > 3) are zero for all the entries; G

_{2}

^{V}= 5.3333, J

_{2}

^{V}= 1.0667.

_{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.

**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.

Molecule | N | G_{1} | G_{2} | G_{3} | G_{4} | G_{5} |
---|---|---|---|---|---|---|

all molecules | 8 | 1.0000 | 6.8889 | 0.4375 | 0.2133 | 0.0625 |

Molecules | J_{1} | J_{2} | J_{3} | J_{4} | J_{5} |
---|---|---|---|---|---|

all molecules | 0.1429 | 0.9841 | 0.0625 | 0.0305 | 0.0089 |

Molecules | G_{1}^{V} | G_{2}^{V} | G_{3}^{V} | G_{4}^{V} | G_{5}^{V} |
---|---|---|---|---|---|

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 |

C_{6}H_{5}–CH=SiH_{2} | 2.5400 | 6.5039 | 0.5297 | 0.3436 | 0.1241 |

C_{6}H_{5}–CH=PH | 1.8800 | 6.6689 | 0.4375 | 0.2611 | 0.0977 |

thiobenzaldehyde | 1.0000 | 6.8889 | 0.4375 | 0.2133 | 0.0625 |

C_{6}H_{5}–CH=GeH_{2} | 2.5400 | 6.5039 | 0.5297 | 0.3436 | 0.1241 |

C_{6}H_{5}–CH=AsH | 2.1000 | 6.6139 | 0.4375 | 0.2886 | 0.1065 |

C_{6}H_{5}–CH=Se | 1.2200 | 6.8339 | 0.4375 | 0.2133 | 0.0713 |

C_{6}H_{5}–CH=SnH_{2} | 2.5400 | 6.5039 | 0.5297 | 0.3436 | 0.1241 |

C_{6}H_{5}–CH=SbH | 2.3200 | 6.5589 | 0.4808 | 0.3161 | 0.1153 |

C_{6}H_{5}–CH=Te | 1.8800 | 6.6689 | 0.4375 | 0.2611 | 0.0977 |

C_{6}H_{5}–CH=PbH_{2} | 2.5400 | 6.5039 | 0.5297 | 0.3436 | 0.1241 |

C_{6}H_{5}–CH=BiH | 2.3200 | 6.5589 | 0.4808 | 0.3161 | 0.1153 |

C_{6}H_{5}–CH=Po | 2.1000 | 6.6139 | 0.4375 | 0.2886 | 0.1065 |

Molecules | J_{1}^{V} | J_{2}^{V} | J_{3}^{V} | J_{4}^{V} | J_{5}^{V} |
---|---|---|---|---|---|

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 |

C_{6}H_{5}–CH=SiH_{2} | 0.3629 | 0.9291 | 0.0757 | 0.0491 | 0.0177 |

C_{6}H_{5}–CH=PH | 0.2686 | 0.9527 | 0.0625 | 0.0373 | 0.0140 |

thiobenzaldehyde | 0.1429 | 0.9841 | 0.0625 | 0.0305 | 0.0089 |

C_{6}H_{5}–CH=GeH_{2} | 0.3629 | 0.9291 | 0.0757 | 0.0491 | 0.0177 |

C_{6}H_{5}–CH=AsH | 0.3000 | 0.9448 | 0.0625 | 0.0412 | 0.0152 |

C_{6}H_{5}–CH=Se | 0.1743 | 0.9763 | 0.0625 | 0.0305 | 0.0102 |

C_{6}H_{5}–CH=SnH_{2} | 0.3629 | 0.9291 | 0.0757 | 0.0491 | 0.0177 |

C_{6}H_{5}–CH=SbH | 0.3314 | 0.9370 | 0.0687 | 0.0452 | 0.0165 |

C_{6}H_{5}–CH=Te | 0.2686 | 0.9527 | 0.0625 | 0.0373 | 0.0140 |

C_{6}H_{5}–CH=PbH_{2} | 0.3629 | 0.9291 | 0.0757 | 0.0491 | 0.0177 |

C_{6}H_{5}–CH=BiH | 0.3314 | 0.9370 | 0.0687 | 0.0452 | 0.0165 |

C_{6}H_{5}–CH=Po | 0.3000 | 0.9448 | 0.0625 | 0.0412 | 0.0152 |

_{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).

_{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.

**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.

_{experiment}and μ

_{vec}

^{V}vary in a similar fashion while μ

_{vec}remains almost constant (μ

_{vec}~0.45D). The electronic effect of the heteroatom (μ

_{vec}

^{V}) dominates over the steric one (μ

_{vec}). In particular, for thiophene (Z=16) μ

_{vec}

^{V}= μ

_{vec}(because χ

_{S}=χ

_{C}). However, the μ

_{vec}

^{V}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 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. - 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., 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}. - 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).

## Acknowledgements

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Torrens, F.
Valence Topological Charge-Transfer Indices for Reflecting Polarity: Correction for Heteromolecules. *Molecules* **2005**, *10*, 334-345.
https://doi.org/10.3390/10020334

**AMA Style**

Torrens F.
Valence Topological Charge-Transfer Indices for Reflecting Polarity: Correction for Heteromolecules. *Molecules*. 2005; 10(2):334-345.
https://doi.org/10.3390/10020334

**Chicago/Turabian Style**

Torrens, F.
2005. "Valence Topological Charge-Transfer Indices for Reflecting Polarity: Correction for Heteromolecules" *Molecules* 10, no. 2: 334-345.
https://doi.org/10.3390/10020334