Quantitative structural relationship between Randic indices, adjacency matrixes, distance matrixes and Dewar resonance energy of linear simple conjugated polyene compounds.

Abstract

Graph theory is a delightful playground for the exploration of proof techniques in Discrete Mathematics and its results have applications in many areas of sciences. Graph theory provides the useful natural mathematical frameworks for the quantitative codification of classical chemical bonding ideas. One of the useful indices for examination of structure-proeprty relationship is Randic index. In this study, is represented the relationship between the Randic index, the determinants of the adjacency matrix and distance matrix to the Dewar resonance energy (DRE) of linear simple conjugated polyenes. The alternative double bonds and conjugation in the polyene compounds are one of the main properties in these compounds. DRE is the quantity which allows one to account for the effects of electron delocalization and thus serves as a measure of the stabilization of a polyenes molecular entity. For calculation the Dewar resonance energy (DRE) of the compounds could use the equation of DRE. The interesting results of concerning among DRE and the above indices are presented.

Keywords: Molecular structure; Molecular topology; Randic index; Polyene Compounds; Dewar resonance energy.

Introduction

Topological indices are the numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Graph theory is a delightful playground for the exploration of proof techniques in Discrete Mathematics and its results have applications in many areas of sciences. Graph theory is a subdiscipline of mathematics that is closely related to both topology and combinatories. A graph is a topological concept rather than a geometrical concept of fixed geometry, and hence Euclidean metric lengths, angles and three-dimensional spatial configurations have no meaning. Chemists employ various types of names and formulas when they wish to communicate information about chemicals and their structures. For the most part names and formulas have no direct, immediate or explicit mathematical meaning. Graph theory provides many different methods of characterizing chemical structures numerically. Graph theory provides many different methods of characterizing chemical structures numerically. Graph theory has been found to be a useful tool in QSAR (Quantitative Structure Activity Relationship) and QSPR (Quantitative Structure Property Relationship). (1-6) Numerous studies have been made relating to the above mentioned fields by using what are called topological indices (TI) (6). One of the stages of topological indices (TI) started when M. Randiu introduced the molecular branching index. (7) In 1975, Randic proposed a topological index that has become one of the most widely used in both QSAR and QSPR studies. Quantitative structure-activity relationships (QSAR) are mathematical models designed for the correlation of various types of biological activity, chemical reactivity, equilibria, physical and physicochemical properties with electronic, steric, hydrophobic and other factors of a molecular structure of a given series of compounds such as substituent constants, topological indices (TI) as well as with solvent and other physicochemical parameters. However, the most important contribution of this stage is probably the great number of applications of TIs in several fields of chemistry. The TIs are based on the original idea of Randic of molecular branching but extended to account for contributions coming from path clusters, clusters and chains of different lengths. (8-15)

Two other important quantities in graph theory are adjacency matrix and distance matrix, (1,16-26) the determinants of which can serve as TIs. The numerical basis for topological indices is provided (depending on how a molecular graph is converted into a numerical value) by either Randic indices, the adjacency matrixes and/or the topological distance matrixes.

In this study, the relationship of Randic index, the determinants of the adjacency matrixes and the distance matrixes, det(A) and det(D), respectively, with the Dewar resonance energy (DRE) will be considered for the linear simple conjugated polyenes (I).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (I)

The compounds I represent initial members of polymerization of acetylene. The alternative double bonds and conjugation in the polyene compounds are one of the main properties in these compounds. One of the properties could be computed using the various types of resonance energy methods is delocalization energy of electrons. For calculating the delocalization energy of compounds (I) was utilized the concepts of the Dewar resonance energy (DRE) and its equation. (See Eq.-7). (27-35)

Mathematical Methods

The Topological Resonance Energy (TRE) scheme rests on the formalism of graph theory. A characteristic polynomial is constructed for the reference structure with the graphs for the given molecule taken into account. The branching index that was introduced by M. Randic is defined as the sum of certain bond contributions calculated from the degree of the bonds suppressed molecular graphs. These bond contributions, named [C.sub.ij] are calculated as:

[C.sub.ij] = [([[delta].sub.i] [[delta].sub.j]).sup.-0.5] (Eq.-1)

where [[delta].sub.i] is the degree of the vertex representing atom "i" , i.e., the number of bonds incident to this atom. Accordingly, the Randic index is defined as (1, 7, 37, 38):

[chi] = [summation] [C.sub.ij] = [summation] [([[delta].sub.i] [[delta].sub.j]).sup.-0.5] (Eq.-2)

Where the summation is carried out over all the bonds of linear simply conjugated polyenes (I). The inverse squared-root of the vertex degree is identified here as a measure of the relative accessible perimeter of an atom from the outside. These perimeters, which have length units, are proposed to be measured in a new unit called the Randiu index ([chi]). On this basis, the bond contributions to the 5DQGLu_ index are relative areas of bond accessibility from the environment.

For two ends of polyene chains I, the Randic indices are: [C.sub.1] = [C.sub.n] = 1/[(2x3).sup.0.5], (only for ethylene: [C.sub.1] = [C.sub.n] = 1/[(2x2).sup.0.5]). For the each carbon atoms between the head and tail in the polyene chains I the Randic indices are: [C.sub.2] = [C.sub.nx -1] = 1/[(3x3).sup.0.5]. On the basis, the final equation for calculating the Randic indices of the linear simple conjugated polyenes (I) is:

[chi] = 2 [1/[(2x3).sup.0.5]] + [([N.sub.B] - 2) / [(3x3).sup.0.5]] (Eq.-3)

and

[chi] = 2 [1/[(6).sup.0.5]] + [(A - 3) / 3] (Eq.-4)

where, "A" is the number of conjugated carbon atoms and "[N.sub.B]" is the number of atoms that the Randic indices are 1/[(3x3).sup.0.5] in the compounds I. If "[n.sub.0]" is equal to "A-3" ([n.sub.0]=A-3), the number of nodes in the HOMO orbitals of the linear simple conjugated polyenes (I) is gained, thus:

[chi] = 0.82 + [0.33 (A-3)] (Eq.-5)

[chi] = 0.82 + [ 0.33 ([n.sub.0])] (Eq.-6)

The adjacency matrixes is defined such that each element [a.sub.ij] equals "1", if and only if atoms "i" and "j" are adjacent (i.e., bonded to each other) while all other [a.sub.ij]'s equal zero. In polyenes I, because of the double bonds between atoms "i" and "j" (adjacent atoms), the [a.sub.ij] equals "2". Some of the adjacency matrixes for polyenes (I) were identified in Fig.-1. (See Table-1).

The distance matrix is defined such that: each element [d.sub.ij] equals the length of the shorter path (i.e., the fewest number of bonds) joining atoms "i" and "j". For polyenes I, some of the distance matrix when: A = 1, 2, 3 were shown in see Fig.-1. (See Table-1).

[FIGURE 1 OMITTED]

The various types of Resonance Energy (RE) methods include the part of the total energy due to electron delocalization. To find the value of the resonance energy (RE), the difference must be calculated between a quantity characterizing experimentally determined energy of a given molecule (such as heat of atomization or heat of formation) and the same characteristic obtained with the aid of an additive scheme, e.g. sum of the bond energies (thermochemical resonance energy). When the total energy or the heat of formation (atomization) is calculated with use of quantum mechanical methods, the RE value is referred to as the Quantum Mechanical Resonance Energy (QMRE). Various schemes for the determination of QMRE are distinguished by the choice of the reference structure which should have non-interacting [pi]-bonds. The Dewar Resonance Energy (DRE) is the quantity which allows one to account for the effects of electron delocalization and thus serves as a measure of the stabilization of a polyenes (I) molecular entity. The model reference structure is not a system of isolated [pi]-bonds (as is the case for "Huckel Resonance Energy" HRE), but a hypothetical polyene with the number of the [pi]- and [omega]-bonds equal to that in a given molecule. The condition of the additively of bond energies for polyenes (I) is adopted. (21-30) For calculation of the Dewar Resonance Energy of compounds (I) could utilized the the Eq.-7 :

DRE = [DELTA][H.sub.f.sup.loc.] - [DELTA][H.sub.f] (Eq.-7)

where [DELTA][H.sub.f.sup.loc.] is the calculated heat of formation of a given conjugated molecule and [DELTA][H.sub.f] is the heat of formation for the reference structure. It is demonstrated that for calculation the heat of formation of the linear simple conjugated polyenes (I) could use from a simple summation function of the number of bonds in (I). (27,30-35,37,38) The Eq.-8 shows :

[DELTA][H.sub.f] = n ([E.sub.C-C]) + [(n +1)([E.sub.C=C])] + [(2n + 4)([E.sub.C-H])] (Eq.-8)

where, "n" is the number of carbon-carbon bonds in polyenes (I) (27-30). But, in this study, the molecular structure of the polyenes (I) have been considered with ab initio methods by using GAUSSIAN-98 computer program that is implemented on a Pentium-PC computer. Firstly, optimizations were performed at 6-31G basis set at the Hartree-Fock (HF) level in the gas phase. To investigate further effect of electron correlation, the geometries were also optimized at the B3LYP/6-31G * level. The results were reverberated in Table-1.

For determining the DRE in polyenes (I) could compute the [DELTA][H.sub.f.sup.loc.] by the use of Eq.-9 (in kcal [mol.sup.-1]) (27):

[DELTA][H.sub.f.sup.loc.] = 7.435 (A) - 0.605([N.sub.H]) (Eq.-9)

Where, "A" and "[N.sub.H]" are the number of carbon and hydrogen atoms in polyenes (I), respectively. The results of these calculations are shown in Table-1.

Discussion

The values of the relative structural coefficients of the polyenes Randic index ([chi]), det(A) and det(D) to the Dewar resonance energy (DRE) of the compounds and their logarithmic data were shown in Table-1. The values shown in table-1 demonstrate that the Randic indices increases with molecular size of polyenes (I). The table reveals the three numerical progression apparent of the Randic indices. On this basis, the distance number of the Randic indices of {[C.sub.4][H.sub.6], [C.sub.10][H.sub.12], [C.sub.16][H.sub.18], [C.sub.22][H.sub.24]}, {[C.sub.6][H.sub.8], [C.sub.12][H.sub.18], [C.sub.18][H.sub.20], [C.sub.24][H.sub.26]} and {[C.sub.8][H.sub.10], [C.sub.14][H.sub.16], [C.sub.20][H.sub.22], [C.sub.26][H.sub.28]} is two (2) units of Randic index. The numbers shown in Fig.-1 (and extended in Table-1 and -2 for larger compounds of the family I) have simple mathematical structure. For example, in lieu of increase each C=C bond in the linear simple conjugated polyenes (I), [DELTA][H.sub.f], [DELTA][H.sub.f.sup.loc.] and DRE were increased 12.25, 13.66 and 1.41 kcal [mol.sup.-1], respectively.(Table-1), the 4, 16, 64, 256 (and so on in Table-1), are even powers of 2; while the numbers 1, 12, 80, 448 (and so on in Table-1) are products of even powers of two and successive odd numbers (i.e., 1, 3, 5, 7 etc). The DRE values increase with the Randic values of the polyenes (I). The values of the determiant of the distance matrixes of the polyenes (I) increase with the number of conjugated double bonds. The values of "log |[D.sub.m]|", were utilized for consideration of the relationship between the values of the relative structural coefficients of polyenes (I). According to the data of Table-2, the logarithmic values of Randic indices, det |A| and det |[D.sub.m]| increase by increasing the number of double bonds of I, while the values log([D.sub.RE]) decrease with increase of the above topological indices.

In Fig.-2 to Fig.-13 were shown (36) two dimensional diagrams of the relationship between the main values of [chi], det |[A.sub.d]|, det|[D.sub.m]|, [DELTA][H.sub.f], [DELTA][H.sub.f.sup.loc.], DRE (see Table-1) and the logarithmic of topological indices (log([chi]), log|[A.sub.d]|, log|[D.sub.m]|) with the values of the log([DELTA][H.sub.f]), log([DELTA][H.sub.f.sup.loc.]) and log([D.sub.RE]) (see Table-2). The Fig.-14 and Fig.-15 are the three dimensional diagrams. The Fig.-14 shows the concerning of log([chi]), log([D.sub.m]) with log([D.sub.RE]). In the Fig.-15 introduced the relationship of log([chi]), log|[A.sub.d]| and log([D.sub.RE]).

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

The Fig.-2 is a plot of the values of "[chi]" versus the "[D.sub.RE]" for polyenes (I). The Fig.-1 shows a high linear correlation between Randic indices and Dewar resonance energy ([D.sub.RE]). The DRE was increased by increasing the double bonds in (I). (See Table-1).

In Fig.-3 could distinguish a curve between logarithm values of Randic indices and log([D.sub.RE]) for polyenes (I). Similarly to Fig.-2, in Fig.-3 could see a high correlation between these logarithmic values. Although, perhaps not obvious from these figures, the relationship between the variables is decidedly nonlinear, which however does not diminish their utility. (See Table-2).

A plot of the "[chi]" versus the [DELTA][H.sub.f] for polyenes (I) was shown in Fig.-4. In Fig.-4 similar to Fig.-2 there is a good correlation between the values.

According to the data of Table-1, the values of Randic indices have a good linear relation with absolute values of [DELTA][H.sub.f.sup.loc.] in the conjugated polyenes (I). The [DELTA][H.sub.f.sup.loc.] was increased by increasing the double bonds in (I). (See Fig.-5).

The high linear relation between logarthmic values of Randic "log([chi])" of the conjugated polyenes (I) and the log([DELTA][H.sub.f]) was shown in the Fig.-6. In lieu of each C=C bond in the polyenes (I) the main values of [chi], [DELTA][H.sub.f], [DELTA][H.sub.f.sup.loc.] and [D.sub.RE] increase (see Table-1).

The Fig.-7 shows the relation between log([chi]) and the log([DELTA][H.sub.f.sup.loc.]). There is a worthless curvature at the first of the curve.

In Fig.-8 the relationship of logarithm values of det|[D.sub.m]| and log([D.sub.RE]) were shown. By increasing the double bonds and log|[D.sub.m]| in I and the values of log([D.sub.RE]) were increased. There is nonlinear concerning among these values.

Similarly to Fig.-8, in Fig.-9 could see a curve between the values of log|[A.sub.d]| and log([D.sub.RE]). Although, perhaps not obvious from these figures, the relationship between the variables is decidedly nonlinear, which however does not diminish their utility.

As could distinguish in Fig.-10 and according to the data of Table-2, the values of log|[D.sub.m]| and the log([DELTA][H.sub.f]) of I show a mild curvature.

Although, in Fig.-11 there is not a linear relationship between the log|[D.sub.m]| and the log([DELTA][H.sub.f.sup.loc.]), but between [C.sub.6][H.sub.8] (A=6)to [C.sub.26][H.sub.28] (A=26) there is linear correlation, comparatively. It is not entirely correlation.

With increasing the number of double bonds in I the log|[A.sub.d]| and log([DELTA][H.sub.f]) increase. The Fig.-12 shows a curve among the values.

The Fig.-13 has very similar shape to Fig.-11. The figure shows the relation between log|[A.sub.d]| and log([DELTA][H.sub.f.sup.loc.]) of the conjugated polyenes (I).

All three Fig.-14 and Fig.-15 show the three dimensional relationship between some of the logarithmic values of "TIs" and [DELTA][H.sub.f], [DELTA][H.sub.f.sup.loc.] and [D.sub.RE]. Three dimensional figure-14 demonstrates comparative relations between log([chi]) and log|[D.sub.m]| and log([D.sub.RE]) in polyenes (I).

In Fig.-15, coud see the near values of log([chi]) and log([D.sub.RE]) as compared with log|[D.sub.m]| and log([D.sub.RE]).

On this basis, the correlation in Fig.-2 to Fig.-6, Fig.-14 and Fig.-15 were better. TIs contain valuable structural information as evidenced by the success of their widespread applications in QSAR and QSPR.

By using and combining of Eq.-4 and Eq.-9, the Eq.-10 was achived. This equation describes correlation of DRE and Randic index ([chi]) for the linear simple conjugated polyene compounds (I).

DRE = 22.305 ([chi]) - [DELTA][H.sub.f] - 0.605 ([N.sub.H]) + 4.015 (Eq.-10)

Where, "DRE", "[chi]", "[DELTA][H.sub.f]" and "[N.sub.H]" are Dewar resonance energy, Randic index, heat of formation for the reference structure, the number hydrogen atoms in polyenes (I), respectively.

Conclusion

The Topological Resonance Energy (TRE) scheme rests on the formalism of graph theory. A characteristic polynomial is constructed for the reference structure with the graphs for the given molecule taken into account. TIs contain valuable structural information as evidenced by the success of their widespread applications in QSAR and QSPR. In this study, the structural relationship between Randic indices, the determinants of the adjacency matrixes and distance matrixes and Dewar resonance energy of linear simple conjugated polyene compounds (I) were presented and disscused. Calculations of the DRE were based on the Dewar resonance energy definitions. In lieu of increase each C=C bond in the linear simple conjugated polyenes (I), [DELTA][H.sub.f], [DELTA][H.sub.f.sup.loc.] and DRE were increased with a good discipline and show very good mathematical structure. The correlation of the Randic index values and log([chi]) with Dewar resonance energy and log([D.sub.RE]) of I resulted in better results than use of other indices for prediction the same of the properties of linear simple conjugated polyenes (I).

Acknowledgment 

The author gratefully acknowledge the colleagues of Chemistry Department of The University of Queensland-Australia for their guidance, advises and useful suggestions.

References

[1] Hansen P. J. and Jurs P.,1988, J. Chem. Edu., 65, 574-580. (and the literature cited therein).

[2] Hosoya H., 1971, Bull. Chem. Soc. Jpn., 44, 2332-2339.

[3] Randic M., 1998, Acta Chim. Slov., 45, 239-252.

[4] Rucker G. and Rucker C., 1999, J. Chem. Inf. Cmput. Sci., 39, 788-802.

[5] Wiener H., 1947, J. Am. Chem. Soc., 17-20.

[6] Du Y. P., Liang Y. Z., Li B. Y. and Xu C. J., 2002, J. Chem. Inf. Cmput. Sci., 42, 1128-1138.

[7] Randic M., 1975, J. Am. Chem. Soc., 97, 6609-6615.

[8] Sabljic A. and Trinajstic N., 1981, Acta Pharm. Ugosl., 31, 189-214.

[9] Sybold P. G., May M. and Bagal U. A., 1987, J. Chem. Edu., 64, 575-582.

[10] Kier L. B. and Hall L. H., 1976, "Molecular Connectivity in Chemistry and Drug Research", Academic Press, New Yourk.

[11] Randic M., 1991, J. Math. Chem., 7, 155-168.

[12] Randic M. and Mills D. and Basak S. C., 2000, Int. J. Quantum Chem., 80, 1199-1209.

[13] Randic M., Plavsic D. and Lers N., 2001, J. Chem. Inf. Cmput. Sci., 41, 657-662.

[14] Randic M. and Basak S. C., 2001, J. Chem. Inf. Cmput. Sci., 41, 614-618.

[15] Randic M. and Pompe M., 2001, J. Chem. Inf. Cmput. Sci., 41, 575-581.

[16] Galvez J., 1998, J. Mol. Struct. (Theochem), 429, 255-264.

[17] Estrada E., 2000, Chem. Phys. Lett., 336, 248-252.

[18] Bonchev D. and Trinajstic N., 1977, J. Chem. Phys., 67, 4517-4533.

[19] Bertz S., 1988, Discrete Applied Math., 19, 65-83.

[20] King R. B., 1983, "Chemical Applications of Topology and Graph Theory", Elsevier, Amesterdam, Netherland.

[21] Ivanciuc O. and Balaban A. T., 1998, Graph Theory in Chemistry; in: The Encyclopedia of Computational Chemistry, Eds. Schleyer P. V. R., Allinger N. L., Clark T., Gasteiger J., Kollman P. A., Schaefer III, H. F., and Schreiner P. R., John Wiley & Sons, Chi Chester.

[22] Ivanciuc O. and Balaban A. T., 1999, The Graph Description of Chemical Structures; in : Topological Indices & Related Descriptors in QSAR and QSPR, Eds. J. Devillers and A. T. Balaban, Gordon & Branch Science Publishers, Amesterdam.

[23] Balaban A., 1982, Highly Discriminating Distance-Based Topological Index, Chem. Phys. Lett., 89, 399-404.

[24] Quin-Nan Hu and Yi-Zeng Liang, 2004, The Branching Number of a Molecular Graph, Internet Electron. J. Mol. Des., 3(6), 335-349.

[25] Barysz M., Plavsic D. and Trinajstic N., 1986, A Note on Topological Indices, Match, 19, 89-116.

[26] Taherpour A. A. and F. Shafeie, 2005, THEOCHEM, "The structural relationship between Randic indices, adjacency matrixes, distance matrixes and maximum wave length of linear simple conjugated polyene compounds", 427, 185-190.

[27] Harris J. M. and Wamser, C. C., 1976, "Fundamentals of Organic Reaction Mechanisms", John Wiley & Sons, New York, USA.

[28] Wheland G. W., 1955,"Resonance in Organic Chemistry", Wiley, New York, USA.

[29] Streitwieser A., Jr., 1962, "Molecular Orbital Theory for Organic Chemists", Wiley, New York, USA.

[30] Dewar M. J. S., 1969, "The Molecular Orbital Theory of Organic Chemistry", McGrow-Hill, New York, USA.

[31] Dewar M. J. S., 1988, "Localization and Delocalization", Eds. Liberman J. F. and Greenberg A., Ch. 1, VCH, New York, USA.

[32] IUPAC Physical Organic Chemistry Glossary, 1999, IUPAC, Pure Appl. Chem., 71(10), 1919-1981.

[33] Baird N. C., 1971, J. Chem. Edu., 48, 509.

[34] Schaad L. J. & Hess B. A., Jr., 1974, J. Chem. Edu., 51, 640.

[35] Hess B. A., Jr., and Schaad L. J., 1971, J. Am. Chem. Soc., 93, 305.

[36] For drawing the graphs of results, we used the Microsoft Office Excel-2003 program.

[37] Kier L. B. and Hall L. H., 2000, J. Chem. Inf. Comput. Sci., 40, 729-795.

[38] Estrada E., 2002, Internet Electron. J. Mol. Des., 1, 360-366. (and the litrature cited therein).

Avat (Arman) Taherpour

Chemistry Department, Graduate Faculty, Islamic Azad University, P. O. Box 38135-567, Arak, Iran. E-mail: [e-mail]

* Sabbatical Address: Chemistry Building, School of Molecular and Microbial Science, The University of Queensland, Qld 4072, Brisbane, Australia.

Table 1: The values of the relative structural coefficients
of polyenes (I).

No. Compounds                  [n.sub.x]                   A

 1  [C.sub.4][H.sub.6]            1                         4
 2  [C.sub.6][H.sub.8]            2                         6
 3  [C.sub.8][H.sub.10]           3                         8
 4  [C.sub.10][H.sub.12]          4                        10
 5  [C.sub.12][H.sub.14]          5                        12
 6  [C.sub.14][H.sub.16]          6                        14
 7  [C.sub.16][H.sub.18]          7                        16
 8  [C.sub.18][H.sub.20]          8                        18
 9  [C.sub.20][H.sub.22]          9                        20
10  [C.sub.22][H.sub.24]         10                        22
11  [C.sub.24][H.sub.26]         11                        24
12  [C.sub.26][H.sub.28]         12                        26

Compounds                        Randic             Adjacency
                             index ([chi])       Det |[A.sub.d]|

[C.sub.4][H.sub.6]                 1.56                    16
[C.sub.6][H.sub.8]                 2.13                    64
[C.sub.8][H.sub.10]                2.89                   256
[C.sub.10][H.sub.12]               3.46                  1024
[C.sub.12][H.sub.14]               4.13                  4096
[C.sub.14][H.sub.16]               4.89                 16384
[C.sub.16][H.sub.18]               5.56                 65536
[C.sub.18][H.sub.20]               6.13                262144
[C.sub.20][H.sub.22]               6.89               1048576
[C.sub.22][H.sub.24]               7.56               4194304
[C.sub.24][H.sub.26]               8.13              16777216
[C.sub.26][H.sub.28]               8.89              27108864

Compounds                       Distance          [DELTA][H.sub.
                            Det |[D.sub.m]|       f.sup.*] (kcal
                                                  [mol.sup.-1])

[C.sub.4][H.sub.6]                    12               24.68
[C.sub.6][H.sub.8]                    80               36.93
[C.sub.8][H.sub.10]                  448               49.18
[C.sub.10][H.sub.12]                2304               61.42
[C.sub.12][H.sub.14]               11264               73.67
[C.sub.14][H.sub.16]               53248               85.92
[C.sub.16][H.sub.18]              245760               98.17
[C.sub.18][H.sub.20]             1114112              110.42
[C.sub.20][H.sub.22]             4980736              122.67
[C.sub.22][H.sub.24]            22020096              234.92
[C.sub.24][H.sub.26]            96468992              147.17
[C.sub.26][H.sub.28]           419430400              159.41

Compounds                   [DELTA][H.sub.f.           DRE
                               sup.loc.]              (kcal
                           (kcal [mol.sup.-1])    [mol.sup.-1])

[C.sub.4][H.sub.6]               26.11                 1.43
[C.sub.6][H.sub.8]               39.77                 2.84
[C.sub.8][H.sub.10]              53.43                 4.25
[C.sub.10][H.sub.12]             67.09                 5.66
[C.sub.12][H.sub.14]             80.75                 7.07
[C.sub.14][H.sub.16]             94.41                 8.48
[C.sub.16][H.sub.18]            108.07                 9,89
[C.sub.18][H.sub.20]            121.73                11.30
[C.sub.20][H.sub.22]            135.39                12.71
[C.sub.22][H.sub.24]            149.05                14.12
[C.sub.24][H.sub.26]            162.71                15.53
[C.sub.26][H.sub.28]            176.37                16.94

* The heats of formation ([DELTA][H.sub.f] in kcal [mol.sup.-1])
were calculated by the use of B3LYP/6-31G*//HF/3-21G methods.

** In lieu of increase each C=C bond in the linear simple conjugated
polyenes (I), [DELTA][H.sub.f], [DELTA].sub.Hf.sup.loc.] and DRE were
increased 12.25,13.66 and 1.41 kcal [mol.sup.-1]. [n.sub.x]= The
number of conjugated double bonds. A= The number of conjugated carbon
atoms.

Table-2: The logarithmic values of the relative structural
coefficients of polyenes (I).

No. Compounds                  [n.sub.x]                A

 1  [C.sub.4][H.sub.6]             1                     4
 2  [C.sub.6][H.sub.8]             2                     6
 3  [C.sub.8][H.sub.10]            3                     8
 4  [C.sub.10][H.sub.12]           4                    10
 5  [C.sub.12][H.sub.14]           5                    12
 6  [C.sub.14][H.sub.16]           6                    14
 7  [C.sub.16][H.sub.18]           7                    16
 8  [C.sub.18][H.sub.20]           8                    18
 9  [C.sub.20][H.sub.22]           9                    20
10  [C.sub.22][H.sub.24]          10                    22
11  [C.sub.24][H.sub.26]          11                    24
12  [C.sub.26][H.sub.28]          12                    26

Compounds                     Log ([chi])         Log|[A.sub.d]|

[C.sub.4][H.sub.6]               0.1931               1.2041
[C.sub.6][H.sub.8]               0.3483               1.8062
[C.sub.8][H.sub.10]              0.4609               2.4082
[C.sub.10][H.sub.12]             0.5514               3.0103
[C.sub.12][H.sub.14]             0.6263               3.6123
[C.sub.14][H.sub.16]             0.6893               4.2144
[C.sub.16][H.sub.18]             0.7451               4.8165
[C.sub.18][H.sub.20]             0.7945               5.4185
[C.sub.20][H.sub.22]             0.8382               6.0206
[C.sub.22][H.sub.24]             0.8785               6.6226
[C.sub.24][H.sub.26]             0.9154               7.2247
[C.sub.26][H.sub.28]             0.9489               7.4331

Compounds                    Log|[D.sub.m]|      Log([DELTA][H.sub.f])

[C.sub.4][H.sub.6]               1.0792               1.393
[C.sub.6][H.sub.8]               1.9031               1.567
[C.sub.8][H.sub.10]              2.6513               1.692
[C.sub.10][H.sub.12]             3.3625               1.788
[C.sub.12][H.sub.14]             4.0517               1.867
[C.sub.14][H.sub.16]             4.7263               1.934
[C.sub.16][H.sub.18]             5.3905               1.992
[C.sub.18][H.sub.20]             6.0469               2.043
[C.sub.20][H.sub.22]             6.6973               2.089
[C.sub.22][H.sub.24]             7.3428               2.130
[C.sub.24][H.sub.26]             7.9844               2.168
[C.sub.26][H.sub.28]             8.6226               2.202

Compounds                  Log([DELTA][H.sub.f.suLog([D.sub.RE])

[C.sub.4][H.sub.6]               1.147                0.155
[C.sub.6][H.sub.8]               1.600                0.453
[C.sub.8][H.sub.10]              1.728                0.628
[C.sub.10][H.sub.12]             1.827                0.753
[C.sub.12][H.sub.14]             1.907                0.849
[C.sub.14][H.sub.16]             1.975                0.928
[C.sub.16][H.sub.18]             2.034                0.995
[C.sub.18][H.sub.20]             2.085                1.053
[C.sub.20][H.sub.22]             2.132                1.104
[C.sub.22][H.sub.24]             2.173                1.150
[C.sub.24][H.sub.26]             2.211                1.191
[C.sub.26][H.sub.28]             2.246                1.229

[n.sub.x]= The number of conjugated double bonds.
A= The number of conjugated carbon atoms