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Article

Valence Topological Charge-Transfer Indices for Dipole Moments

by
Francisco Torrens
Institut Universitari de Ciència Molecular, Universitat de València, Dr. Moliner 50, E-46100 Burjassot, València, Spain
Molecules 2003, 8(1), 169-185; https://doi.org/10.3390/80100169
Submission received: 1 January 2003 / Published: 31 January 2003
(This article belongs to the Special Issue Selected Papers from the International Symposium on Frontiers in Molecular Science (ISFMS 2002))
Browse Figures
Versions Notes

Introduction

The rationale for transdermal drug delivery needs to be carefully identified [1]. There has been great interest in developing systems for controlled delivery of drugs and other bioactive substances [2]. One technique is to attach a specialized patch on the skin with uncoated polymer matrices containing the embedded drug [3]. Other technique involves covering most of the matrix with an impermeable material [4]. The pathways that exist in a porous membrane used to deliver drugs over a continuous period form a percolating path with usual fractal structures [5,6].
Percutaneous penetration depends essentially on the lipophilicity of the substances tested [7]. The use of homologous series of compounds made it possible to establish behaviour models that allow predicting the percutaneous absorption capacity of chemically related substances [8,9]. Much of the theoretical background information on percutaneous absorption was developed from studies of nonelectrolytic permeating species [10,11]. However, the vehicle’s pH may have a profound influence upon the percutaneous delivery from topical products [12]. Depending on the pKa of the compound and on the pH of the vehicle, an equilibrium mixture of ionized and unionized species would be present in the immediate vicinity of the skin [13]. The permeation of ionized drugs through the skin must be taken into account [14]. Penetration enhancers may act by either interacting with the highly ordered lipid structure or modifying the partitioning of the drug into the tissue [15,16].
Different phenyl alcohols are included in many pharmaceutical and cosmetic products, e.g., bacteriostatic agents and essential oils. Benzyl alcohol, which is used as a bacteriostatic agent and also as a solvent, was evaluated as a percutaneous enhancer [17]. López et al. assessed other phenyl alcohols as enhancers for 5-fluorouracil [8], which was used in in vitro diffusion experiments [18]. Other authors employed dipalmitoylphosphatidylcholine as enhancer for azone [19].
In previous papers, molecular topological charge-transfer indices were applied to the calculation of the molecular dipole moment of hydrocarbons [20]. New valence charge-transfer indices were introduced to take into account the presence of heteroatoms in the molecules and applied to valence-isoelectronic series of benzene and styrene [21]. In this paper, the method has been applied to the calculation of the dipole moment of a homologous series of phenyl alcohols. In the next section, the topological charge-transfer indices are revised. Next, the valence charge-transfer indices for heteroatoms are presented. Following that, the calculation results are discussed. The last section summarizes my conclusions.

Topological charge-transfer indices

The two most important matrices that delineate the labelled chemical graph are the adjacency ( Molecules 08 00169 i003) [22,23,24] and the distance ( Molecules 08 00169 i004) matrices, wherein Dij = Iij if i=j, “0” otherwise; lij is the shortest edge count between vertices i and j [25]. In Molecules 08 00169 i003, Aij = 1 if vertices i and j are adjacent, “0” otherwise. The Molecules 08 00169 i005 matrix is the matrix whose elements are the squares of the reciprocal distances Molecules 08 00169 i006 [26]. Now, the intermediate matrix Molecules 08 00169 i007 is defined as the matrix product of Molecules 08 00169 i003 by Molecules 08 00169 i005:
Molecules 08 00169 i008
The charge-transfer matrix Molecules 08 00169 i009 is defined as Molecules 08 00169 i010, where Molecules 08 00169 i011 is the transpose of Molecules 08 00169 i007 [27]. By agreement, Cii = Mii. For i ≠ j, the Cij terms represent a measure of the intramolecular net charge transferred from atom j to i.
Gálvez et al. defined the topological charge-transfer indices Gk as the sum, in absolute value, of the Cij terms defined for the vertices i,j placed at a topological distance Dij equal to k:
Molecules 08 00169 i012
where N is the number of vertices in the graph, Dij are the entries of the Molecules 08 00169 i004 matrix, and δ is the Kronecker δ, being δ = 1 for i = j and δ = 0 for i ≠ j. Gk represents the sum of all the Cij terms, for every pair of vertices i and j at topological distance k. Gálvez et al. also introduced other topological charge-transfer index, Jk, as
Molecules 08 00169 i013
This index represents the mean value of the charge transfer for each edge, since the number of edges for acyclic compounds is N-1. The Gk and Jk indices were applied to the calculation of the molecular dipole moment of hydrocarbons, the boiling temperature of hydrocarbons and alcohols, the vaporization enthalpy of alkanes, the activity of drugs [28,29,30], the capacity and separation factors in chromatographic chiral separations of biomolecules [31], the resonance energy of hydrocarbons, the log IC50 of the D2 dopamine receptor and the σ receptor of piperidines and the sedative character of chiral barbiturates [32].
The algebraic semisum charge-transfer index μalg is defined as
Molecules 08 00169 i014
where Ce is the Cij index for vertices i and j connected by edge e. In the first equation, the sum extends for all pairs of adjacent vertices in the molecular graph. In the second expression, the sum extends for the m edges in the molecular graph. Therefore, μalg is intended as a measure of the molecular dipole moment calculated by algebraic semisum of Ce.
An edge-to-edge analysis of μ suggests that each edge dipole moment μe connecting vertices i and j can be evaluated from the corresponding edge Ce index as
Molecules 08 00169 i015
Each edge dipole can be associated with a vector Molecules 08 00169 i016 in space. This vector has magnitude μe, lies in the edge e connecting vertices i and j, and its direction is from j to i. The molecular dipole moment vector results the vector sum of the edge dipole moments as
Molecules 08 00169 i017
summed for all the m edges in the molecular graph. The vector semisum charge-transfer index μvec is defined as the norm of Molecules 08 00169 i018:
Molecules 08 00169 i019
Therefore, μvec is intended as a measure of the molecular dipole moment calculated by vector semisum of edge Molecules 08 00169 i020.
As μvec is associated with a vector in space, the three-dimensional structure of the molecule is needed. The method followed to get it is outlined as follows.
1.
Read the Cartesian coordinates of the atoms.
2.
Determine which atoms are bonded to which other atoms. The distance between two atoms is calculated and, if it is less than a certain value, a bond is assumed to exist between the two atoms.
3.
Build the charge transfer terms Cij.
4.
Calculate the vector semisum dipole moment μvec.
Therefore, μvec is not a pure topological index. It is rather a linear combination of topological indices, but the coefficients are geometric descriptors.

Valence charge-transfer indices for heteroatoms

In valence charge-transfer 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 Molecules 08 00169 i003. For each heteroatom X, its entry Aii is redefined as
Molecules 08 00169 i021
to give a modified Molecules 08 00169 i022 matrix, where xx and xc are the electronegativities of heteroatom X and carbon, respectively, in Pauling units. Notice that the subtractive term keeps Molecules 08 00169 i023 for the carbon atom (Equation 5). Moreover, the multiplicative factor reproduces Molecules 08 00169 i024 for oxygen, which was taken as standard in previous works. From the modified Molecules 08 00169 i022 matrix, the valence Molecules 08 00169 i025 and Molecules 08 00169 i026 matrices, Molecules 08 00169 i027, Molecules 08 00169 i028 and topological charge-transfer indices Molecules 08 00169 i001 and Molecules 08 00169 i002 can be calculated by following the former procedure with the Molecules 08 00169 i022 matrix. The Molecules 08 00169 i029, Molecules 08 00169 i001 and Molecules 08 00169 i002 descriptors are graph invariants. The main difference between μvec and Molecules 08 00169 i028 is that μvec is sensitive only to the steric effect of the heteroatoms while is sensitive to both electronic and steric effects.

Calculation results and discussion

The molecular charge-transfer indices Gk, Jk, Molecules 08 00169 i001 and Molecules 08 00169 i002 (with k < 6) for a series of 13 phenyl alcohols (8 form a homologous series and 5 are congeneric) are reported in Table 1. In the homologous series, G1 and G2 are sensitive to the presence of the alkyl chain. G3, G4 and G5 indicate the presence of at least 2, 3 and 4 carbon atoms in the alkyl chain, respectively. Molecules 08 00169 i030 is influenced by the presence of the alkyl chain. Molecules 08 00169 i031 – Molecules 08 00169 i034 point out the presence of at least 2–5 carbon atoms in the alkyl chain.
Table
Table 1. Values of the Gk and Jk charge-transfer indices up to the fifth order for a homologous series of phenyl alcohols.
Table 1. Values of the Gk and Jk charge-transfer indices up to the fifth order for a homologous series of phenyl alcohols.
MoleculeNG1G2G3G4G5
Phenol 72.00000.88890.37500.22220.0000
benzyl alcohol 81.25006.77780.81250.41330.1250
2-phenyl-1-ethanol 91.25006.77780.87500.56440.2431
3-phenyl-1-propanol101.25006.77780.87500.60440.3333
4-phenyl-1-butanol111.25006.77780.87500.60440.3611
5-phenyl-1-pentanol121.25006.77780.87500.60440.3611
6-phenyl-1-hexanol131.25006.77780.87500.60440.3611
7-phenyl-1-heptanol141.25006.77780.87500.60440.3611
1-phenyl-2-propanol102.25006.77780.93750.71560.3611
2-phenyl-2-propanol102.75007.22221.56250.79560.3750
3-phenyl-2-propen-1-ol101.25008.22220.87500.48440.2708
1-phenyl-1-pentanol121.75007.11111.31250.87560.4861
1-phenyl-2-pentanol122.25007.00001.06250.75560.4792
MoleculeJ1J2J3J4J5
phenol0.33330.14810.06250.03700.0000
benzyl alcohol0.17860.96830.11610.05900.0179
2-phenyl-1-ethanol0.15630.84720.10940.07060.0304
3-phenyl-1-propanol0.13890.75310.09720.06720.0370
4-phenyl-1-butanol0.12500.67780.08750.06040.0361
5-phenyl-1-pentanol0.11360.61620.07950.05490.0328
6-phenyl-1-hexanol0.10420.56480.07290.05040.0301
7-phenyl-1-heptanol0.09620.52140.06730.04650.0278
1-phenyl-2-propanol0.25000.75310.10420.07950.0401
2-phenyl-2-propanol0.30560.80250.17360.08840.0417
3-phenyl-2-propen-1-ol0.13890.91360.09720.05380.0301
1-phenyl-1-pentanol0.15910.64650.11930.07960.0442
Molecule Molecules 08 00169 i030 Molecules 08 00169 i031 Molecules 08 00169 i032 Molecules 08 00169 i033 Molecules 08 00169 i034
phenol2.20001.10000.36390.08470.0000
benzyl alcohol2.95006.66110.56250.25330.0370
2-phenyl-1-ethanol2.95007.10560.74440.40440.1319
3-phenyl-1-propanol2.95007.10560.99440.50190.2222
4-phenyl-1-butanol2.95007.10560.99440.66190.2824
5-phenyl-1-pentanol2.95007.10560.99440.66190.3936
6-phenyl-1-hexanol2.95007.10560.99440.66190.3936
7-phenyl-1-heptanol2.95007.10560.99440.66190.3936
1-phenyl-2-propanol3.45007.65560.80690.55560.2500
2-phenyl-2-propanol3.45008.20561.31250.63560.2870
3-phenyl-2-propen-1-ol2.95008.32780.63060.38190.1597
1-phenyl-1-pentanol2.95007.32221.18190.77310.4861
Molecule Molecules 08 00169 i035 Molecules 08 00169 i036 Molecules 08 00169 i037 Molecules 08 00169 i038 Molecules 08 00169 i039
phenol0.36370.18330.06060.01410.0000
benzyl alcohol0.42140.95160.08040.03620.0053
2-phenyl-1-ethanol0.36880.88820.09310.05060.0165
3-phenyl-1-propanol0.32780.78950.11050.05580.0247
4-phenyl-1-butanol0.29500.71060.09940.06620.0282
5-phenyl-1-pentanol0.26820.64600.09040.06020.0358
6-phenyl-1-hexanol0.24580.59210.08290.05520.0328
7-phenyl-1-heptanol0.22690.54660.07650.05090.0303
1-phenyl-2-propanol0.38330.85060.08970.06170.0278
2-phenyl-2-propanol0.38330.91170.14580.07060.0319
3-phenyl-2-propen-1-ol0.32780.92530.07010.04240.0177
1-phenyl-1-pentanol0.26820.66570.10740.07030.0442
1-phenyl-2-pentanol0.31360.69600.09560.06660.0335
The molecular dipole moments μ (experimental and calculated as vector semisums of Cij or Molecules 08 00169 i029) are listed in Table 2. As experimental values are not available for all the series, some reference values are calculated with program MOPAC-AM1. The reliability of the results has been tested with the first four entries in Table 2, for which experimental data are available. AM1 calculations adequately reproduce the oscillatory behaviour of the experimental data, mimicking two minima for phenol and 2-phenyl-1-ethanol and two maxima for benzyl alcohol and 3-phenyl-1-propanol. This test suggests that AM1 allows a good approximation, at least for the general performance of the homologous series as a whole, and that the error is sufficiently constant throughout the homologous series. The number of compounds in the homologous series has not been increased because longer phenyl alcohols are not percutaneous enhancers owing to their lower transdermal penetration. In particular, 3-phenyl-2-propen-1-ol has been chosen in order to compare the influence of a double bound in the alkyl chain region of a phenyl alcohol. The experimental μ decreases ca. 3% from that of 3-phenyl-1-propanol due to the presence of the double bond. The presence of an enol group (conjugated double bond in β position with respect to the -OH group) lends 3-phenyl-2-propen-1-ol to particular structural characteristics (greater rigidity than for 3-phenyl-1-propanol) with a lower μ. For the three phenyl propanols, μexperiment decreases 5% from 3-phenyl-1-propanol (primary alcohol) to 1-phenyl-2-propanol (secondary alcohol) and 11% to 2-phenyl-2-propanol (tertiary alcohol). For the three phenyl pentanols, μexperiment increases 7% from 5-phenyl-1-pentanol (primary alcohol) to 1-phenyl-1-pentanol (secondary alcohol with both –phenyl and –OH groups in terminal position) and decreases 8% to 1-phenyl-2-pentanol (secondary alcohol).
Table
Table 2. Molecular dipole moment values, μ (D), for a series of phenyl alcohols with charge-transfer indices.
Table 2. Molecular dipole moment values, μ (D), for a series of phenyl alcohols with charge-transfer indices.
MoleculeNumber of carbon atoms in alkyl chainVector semisumValence vector semisumExperimenta
phenol00.7372.4311.400 (1.233b)
benzyl alcohol10.5892.4871.700 (1.568b)
2-phenyl-1-ethanol20.7002.2571.590 (1.497b)
3-phenyl-1-propanol30.5732.5191.640 (1.597b)
4-phenyl-1-butanol40.7022.2491.345b
5-phenyl-1-pentanol50.5732.5191.626b
6-phenyl-1-hexanol60.7022.2501.346b
7-phenyl-1-heptanol70.5732.5181.634b
1-phenyl-2-propanol30.9232.3471.564b
2-phenyl-2-propanol30.7802.8211.463b
3-phenyl-2-propen-1-ol30.5612.4951.591b
1-phenyl-1-pentanol50.4262.7941.746b
1-phenyl-2-pentanol50.8952.3671.496b
a Taken from Reference 60.b Calculations carried out with program MOPAC-AM1.
Figure 1 illustrates how the dipole moment of the homologous phenyl alcohols fluctuates with the number of C atoms in the alkyl chain, n. Experimental μ and calculated Molecules 08 00169 i028 vary in an alternate fashion: for odd n, μ is greater than for even n. However, μvec presents the opposite tendency. As μvec is not sensitive to the electronic effect of the oxygen atom, it is clear that the steric (μvec) and electronic ( Molecules 08 00169 i028) factors are antagonistic, and that the electronic effect dominates over the steric one. The μexperiment decreases ca. 4% either from to 6 or from to 7.
Molecules 08 00169 g001 550
Figure 1. Dipole moment of homologous phenyl alcohols as a function of the number of C atoms in alkyl chain.
Figure 1. Dipole moment of homologous phenyl alcohols as a function of the number of C atoms in alkyl chain.
Molecules 08 00169 g001
The experimental dipole moments and charge-transfer indices have been correlated for the homologous series. The best linear regression results:
Molecules 08 00169 i040
The mean absolute percentage error (MAPE) is 4.53% and the approximation error variance (AEV) is 0.5087. All other models with greater MAPE and AEV have been discarded. The presence of charge-transfer (G3) and valence ( Molecules 08 00169 i038) indices is representative of charge-transfer processes and clearly conditions the polar character of the correlated compounds. The best non-linear model for μ does not improve the results.
If the congeneric set is included in the model of phenyl alcohols, the best linear regression for all μ results:
Molecules 08 00169 i041
and AEV increases 21%. The best non-linear model for μ does not improve the results.
If all μ are calculated with AM1, the best linear regression results:
Molecules 08 00169 i042
and AEV augments 12%. The best non-linear model for μ does not improve the results.
If μvec is included in the model of phenyl alcohols, the best linear regression for μ results:
Molecules 08 00169 i043
and AEV decreases 72%. Notice that the negative sign of the μvec coefficient is due to the opposite trend of μvec and μexperiment. The best non-linear model for μ results:
Molecules 08 00169 i044
and AEV diminishes 86%.
If Molecules 08 00169 i028 is included in the model, the best linear regression for μ results:
Molecules 08 00169 i045
and AEV drops 73%. The non-linear model with Molecules 08 00169 i028 results:
Molecules 08 00169 i046
and AEV decreases 92%. Due to the complexity of Equations 8 and 10, a Fortran program has been written to calculate μ. The program is available from the author at Internet (Francisco.Torrens@uv.es).
No superposition of the corresponding Gk–Jk and Molecules 08 00169 i001– Molecules 08 00169 i002 pairs is observed in Equations 6–10. This diminishes the risk of co-linearity in the fit, given the close relationship between each pair Gk, Jk in Equation 2 [33].
The correlation coefficient found between cross-validated representatives and the property values Rcv has been calculated with the leave-n-out procedure for Equations 6–10 [34]. The procedure furnishes a new method for selecting the best set of descriptors according to the criterion of maximization of the value of Rcv. The Rcv calculations for a homologous series of phenyl alcohols are given in Table 3 for 1 ≤ n ≤ 5. In general, Rcv decreases with n. In particular, Equations 9-10 give greater Rcv than Equations 7-8 (2%) and than Equation 6 (86%) for the whole range of n given in Table 3, although the results for Equations 7 and 9 are rather similar. The corresponding interpretation is that Equations 9-10 are more predictive than Equations 7-8 or 6.
Table
Table 3. Cross-validation correlation coefficient in a leave-n-out procedure for a homologous series of phenyl alcohols.
Table 3. Cross-validation correlation coefficient in a leave-n-out procedure for a homologous series of phenyl alcohols.
nEquation 6Equation 7Equation 8Equation 9Equation 10
10.4770.8340.9150.8230.956
2–0.8270.9160.8210.954
3–0.8190.9160.8210.951
4–0.8060.9150.8200.935
5––0.8900.823-0.951
In the rest of this section, new parameters are calculated to illustrate the importance of linear and non-linear models in pharmacology. Another intramolecular property related with charge-transfer is the quadrupole moment tensor Θ, defined by
Molecules 08 00169 i047
where Molecules 08 00169 i048 is the i-th element of charge at the point Molecules 08 00169 i049 relative to an origin fixed at the centre of mass in the molecule [35]. The δab stands for the Kronecker δ. The subscripts a, b… denote vector or tensor components and can be equal to the Cartesian components x, y, z. Tensor Θ is calculated with our program POLAR [36,37]. The fractal dimension D of the solvent-accessible surface As of the phenyl alcohols may then be obtained according to Lewis and Rees as
Molecules 08 00169 i050
where R is the radius of a probe representing the solvent [38]. It is calculated with our program TOPO. The fractal dimension D provides a quantitative indication of the degree of surface accessibility toward different solvents [39,40,41,42,43,44,45,46]. In previous papers, the fractal dimension of transdermal-delivery drug models (phenyl alcohols [47] and 4-alkylanilines [48]) was calculated. A linear model for the molecular quadrupole moment of the homologous phenyl alcohols vs. fractal dimension is shown in Figure 2.
Molecules 08 00169 g002 550
Figure 2. Quadrupole moment of homologous phenyl alcohols vs. the fractal dimension.
Figure 2. Quadrupole moment of homologous phenyl alcohols vs. the fractal dimension.
Molecules 08 00169 g002
The 1-octanol–water partition coefficient of a substance soluble in 1-octanol and water is defined as
Molecules 08 00169 i051
where a1-octanol and awater are the activities of the solute in 1-octanol and water, respectively [49]. P is calculated with the equation:
Molecules 08 00169 i052
at a given T, which is taken as 298K. R is the gas constant, and Molecules 08 00169 i053 and Molecules 08 00169 i054 (in kJ·mol-1) are the standard-state free energies of solvation of a given solute considered in 1-octanol and water, respectively. The solvation energies are calculated with our program SCAP [50,51,52,53,54,55]. A non-linear model for of the homologous phenyl alcohols vs. fractal dimension is illustrated in Figure 3.
Molecules 08 00169 g003 550
Figure 3. 1-Octanol–water partition coefficient of homologous phenyl alcohols vs. the fractal dimension.
Figure 3. 1-Octanol–water partition coefficient of homologous phenyl alcohols vs. the fractal dimension.
Molecules 08 00169 g003
The molar concentration necessary to produce a 1:1 complex with bovine serum albumin (BSA) via equilibrium dialysis, C, provides information on the binding and conformational perturbation of macromolecules by smaller compounds [56]. It is calculated with SCAP. A non-linear model of for the homologous phenyl alcohols vs. fractal dimension is illustrated in Figure 4.
Molecules 08 00169 g004 550
Figure 4. Molar concentration of homologous phenyl alcohols necessary to produce a 1:1 complex with BSA via equilibrium dialysis vs. the fractal dimension.
Figure 4. Molar concentration of homologous phenyl alcohols necessary to produce a 1:1 complex with BSA via equilibrium dialysis vs. the fractal dimension.
Molecules 08 00169 g004
The hydrophile–lipophile balance (HLB) is a scale that characterizes surfactants in emulsions as solubilizers, emulsifiers, wetting agents or antifoaming agents [57,58]. It is calculated with SCAP. A non-linear model of HLB for the homologous phenyl alcohols vs. fractal dimension is shown in Figure 5.
Molecules 08 00169 g005 550
Figure 5. Hydrophile–lipophile balance of homologous phenyl alcohols vs. the fractal dimension.
Figure 5. Hydrophile–lipophile balance of homologous phenyl alcohols vs. the fractal dimension.
Molecules 08 00169 g005
The best linear model for the fractal dimension of the phenyl alcohols vs. a series of physicochemical parameters results:
Molecules 08 00169 i055
where MW is the molecular weight, G is the molecular globularity and G’ is the molecular rugosity. Index G is calculated as G = Se/S, where Se is the surface area of a sphere whose volume is equal to the molecular volume V, and S is the molecular surface area. Index G’ is calculated as G’ = S/V. All the descriptors are calculated with TOPO. Notice that Equation 15 contains five parameters versus only eight data, allowing three degrees of freedom. The statistical significance of the fit is that the error introduced by fitting is much lower than the error due to the individual points and can be disregarded. Therefore, the data points can be replaced by the fit without adding an appreciable error. The very question is the predictive power of the fit for points out of the fitting set [59]. However, it is not the case because longer phenyl alcohols in the aromatic series are not percutaneous enhancers owing to their lower transdermal penetration. On the other hand, the best non-linear model for the fractal dimension does not improve the results.

Conclusions

From the preceding results the following conclusions can be drawn:
1
Inclusion of the oxygen atom in the π-electron system is beneficial for the description of the dipole moment, owing to either the role of the additional p 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 oxygen atom shows that the two factors are antagonistic, and that the electronic factor dominates over the steric one.
2
In 3-phenyl-2-propen-1-ol, the double bond in the side chain lends to a more rigid structure with a lower dipole moment.
3
Linear and non-linear correlation models have been obtained for the molecular dipole moments of phenyl alcohols. The new μvec and Molecules 08 00169 i028 charge-transfer indices have improved the multivariable regression equations for μ, diminishing the risk of co-linearity in the fit.
4
The linear correlation between D and Θ, and various non-linear correlations between D, log P, log 1/C and HLB point not only to a homogeneous molecular structure of the phenyl alcohols but also to the ability to predict and tailor drug properties. The latter is nontrivial in pharmacology.
Work is in progress on the calculation of the dipole moments of a homologous series of 4-alkylanilines, which are also percutaneous enhancers.

Acknowledgements

I wish to thank Dr. E. Besalú for providing me several versions of his full linear leave-many-out program prior to publication, and Drs. Gálvez and de Julián-Ortiz for their kind comments. The author acknowledges financial support of the Spanish MCT (Plan Nacional I+D+I, Project No. BQU2001-2935-C02-01).

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Torrens, F. Valence Topological Charge-Transfer Indices for Dipole Moments. Molecules 2003, 8, 169-185. https://doi.org/10.3390/80100169

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Torrens F. Valence Topological Charge-Transfer Indices for Dipole Moments. Molecules. 2003; 8(1):169-185. https://doi.org/10.3390/80100169

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Torrens, Francisco. 2003. "Valence Topological Charge-Transfer Indices for Dipole Moments" Molecules 8, no. 1: 169-185. https://doi.org/10.3390/80100169

APA Style

Torrens, F. (2003). Valence Topological Charge-Transfer Indices for Dipole Moments. Molecules, 8(1), 169-185. https://doi.org/10.3390/80100169

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