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

Abstract

Graph theory graph theory

Mathematical theory of networks. A graph consists of vertices (also called points or nodes) and edges (lines) connecting certain pairs of vertices. An edge that connects a node to itself is called a loop.
is a delightful playground for the exploration of proof techniques in Discrete Mathematics Discrete mathematics, also called finite mathematics or Decision Maths, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity.  and its results have applications in many areas of sciences. Graph theory provides the useful natural mathematical frameworks for the quantitative codification The collection and systematic arrangement, usually by subject, of the laws of a state or country, or the statutory provisions, rules, and regulations that govern a specific area or subject of law or practice.  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 In mathematics and computer science, the adjacency matrix of a finite directed or undirected graph G on n vertices is the n × n matrix where the nondiagonal entry  and distance matrix to the Dewar resonance energy (DRE DRE
Digital rectal examination.

Mentioned in: Rectal Examination
) of linear simple conjugated conjugated
Conjugate.

estrogens, conjugated Warning - Hazardous drug!

C.E.S.
polyenes. The alternative double bonds and conjugation conjugation, in genetics
conjugation, in genetics: see recombination.
conjugation, in grammar
conjugation: see inflection.
in the polyene polyene /pol·y·ene/ (pol´e-en)
1. a chemical compound with a carbon chain of four or more atoms and several conjugated double bonds.

2. any of a group of antifungal antibiotics with such a structure (e.g.
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 topology, branch of mathematics, formerly known as analysis situs, that studies patterns of geometric figures involving position and relative position without regard to size. ; Randic index; Polyene Compounds; Dewar resonance energy.

Introduction

Topological to·pol·o·gy
n. pl. to·pol·o·gies
1. Topographic study of a given place, especially the history of a region as indicated by its topography.

2.
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 sub·dis·ci·pline
n.
A field of specialized study within a broader discipline; a subfield.
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 QSAR Quantitative Structure-Activity Relationship
QSAR Quality System Audit Report
QSAR Quality Service Activity Report
QSAR Québec Secours Search and Rescue (Canada)
(Quantitative Structure Activity Relationship) and QSPR QSPR Quantitative Structure-Property Relationship
QSPR Quarterly Statistical Performance Report
(Quantitative Structure Property Relationship). (1-6) Numerous studies have been made relating to relating to relate prepconcernant

relating to relate prepbezüglich +gen, mit Bezug auf +acc
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 Quantitative structure-activity relationship (QSAR) is the process by which chemical structure is quantitatively correlated with a well defined process, such as biological activity or chemical reactivity.  (QSAR) are mathematical models
Note: The term model has a different meaning in model theory, a branch of mathematical logic. An artifact which is used to illustrate a mathematical idea is also called a mathematical model and this usage is the reverse of the sense explained below.
designed for the correlation of various types of biological activity, chemical reactivity, equilibria, physical and physicochemical physicochemical /phys·i·co·chem·i·cal/ (fiz?i-ko-kem´ik-il) pertaining to both physics and chemistry.

phys·i·co·chem·i·cal
1. Relating to both physical and chemical properties.
properties with electronic, steric steric /ste·ric/ (ster´ik) pertaining to the arrangement of atoms in space; pertaining to stereochemistry.

ster·ic or ster·i·cal
n.
, hydrophobic hydrophobic /hy·dro·pho·bic/ (-fo´bik)
1. pertaining to hydrophobia (rabies).

3.
and other factors of a molecular structure of a given series of compounds such as substituent substituent /sub·stit·u·ent/ (-stich´u-ent)
1. a substitute; especially an atom, radical, or group substituted for another in a compound.

2. of or pertaining to such an atom, radical, or group.
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 A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .] (I)

The compounds I represent initial members of polymerization polymerization

Any process in which monomers combine chemically to produce a polymer. The monomer molecules—which in the polymer usually number from at least 100 to many thousands—may or may not all be the same.
of acetylene acetylene (əsĕt`əlēn') or ethyne (ĕth`īn), HC≡CH, a colorless gas. It melts at −80.8°C; and boils at −84.0°C;. . 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 TRE Tampere (Finland)
TRE Tribunal Regional Eleitoral (Brazil)
TRE Trinity Railway Express (Texas)
TRE Theologische Realenzyklopädie
) scheme rests on the formalism Formalism
or Russian Formalism

Russian school of literary criticism that flourished from 1914 to 1928. Making use of the linguistic theories of Ferdinand de Saussure, Formalists were concerned with what technical devices make a literary text literary, apart

In linear algebra, one associates a polynomial to every square matrix, its 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 A corner point of a triangle or other geometric image. Vertices is the plural form of this term. See vertex shader.  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 summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) ] [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 (mathematics) inverse - Given a function, f : D -> C, a function g : C -> D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold.  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 ethylene (ĕth`əlēn') or ethene (ĕth`ēn), H2C=CH2, a gaseous unsaturated hydrocarbon. It is the simplest alkene. : [C.sub.1] = [C.sub.n] = 1/[(2x2).sup.0.5]). For the each carbon atoms Noun 1. carbon atom - an atom of carbon
atom - (physics and chemistry) the smallest component of an element having the chemical properties of the element
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 orbitals (ōrˑ·b·t  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 Atomization

The process whereby a bulk liquid is transformed into a multiplicity of small drops. This transformation, often called primary atomization, proceeds through the formation of disturbances on the surface of the bulk liquid, followed by their
or heat of formation) and the same characteristic obtained with the aid of an additive additive

In foods, any of various chemical substances added to produce desirable effects. Additives include such substances as artificial or natural colourings and flavourings; stabilizers, emulsifiers, and thickeners; preservatives and humectants (moisture-retainers); and
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 HRE
abbr.
Holy Roman Empire
), 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 C-C Carbon-Carbon
C-C Carotid-Cavernous (relating to the carotid artery and the sinuses)
]) + [(n +1)([E.sub.C=C])] + [(2n + 4)([E.sub.C-H])] (Eq.-8)

where, "n" is the number of carbon-carbon bonds A carbon-carbon bond is a covalent bond between two carbon atoms. The most common form is the single bond – a bond composed of two electrons, one from each of the two atoms.  in polyenes (I) (27-30). But, in this study, the molecular structure of the polyenes (I) have been considered with ab initio [Latin, From the beginning; from the first act; from the inception.] An agreement is said to be "void ab initio" if it has at no time had any legal validity.  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 logarithmic

pertaining to logarithm.

logarithmic relationship
when the logs of two variables plotted against each other create a straight line.
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 In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance. . For example, in lieu of Instead of; in place of; in substitution of. It does not mean in addition to.  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 conjugated double bond
n.
Two double bonds in a compound that are separated by a single bond.
. 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 according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
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 logarithm (lŏg`ərĭthəm) [Gr.,=relation number], number associated with a positive number, being the power to which a third number, called the base, must be raised in order to obtain the given positive number.  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 A system in which the output is not a uniform relationship to the input.

nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input.
, 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 curvature

Measure of the rate of change of direction of a curved line or surface at any point. In general, it is the reciprocal of the radius of the circle or sphere of best fit to the curve or surface at that point.
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 acknowledgment, in law, formal declaration or admission by a person who executed an instrument (e.g., a will or a deed) that the instrument is his. The acknowledgment is made before a court, a notary public, or any other authorized person.

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

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of Computational Chemistry Computational chemistry is a branch of chemistry that uses computers to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids. , 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 John Wiley may refer to:
• John Wiley & Sons, publishing company
• John C. Wiley, American ambassador
• John D. Wiley, Chancellor of the University of Wisconsin-Madison
• John M. Wiley (1846–1912), U.S.
& Sons, Chi Chester.

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[23] Balaban A., 1982, Highly Discriminating dis·crim·i·nat·ing
1.
a. Able to recognize or draw fine distinctions; perceptive.

b. Showing careful judgment or fine taste:
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[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 New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of
, USA.

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

[29] Streitwieser A., Jr., 1962, "Molecular Orbital Theory molecular orbital theory, detailed explanation of how electrons are distributed in stable molecules. In the simpler valence theory of the chemical bond, each atom in a molecule is assumed to retain its own electrons.  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 Customizing software and documentation for a particular country. It includes the translation of menus and messages into the native spoken language as well as changes in the user interface to accommodate different alphabets and culture. See internationalization and l10n.  and Delocalization", Eds. Liberman J. F. and Greenberg A., Ch. 1, VCH VCH Victoria County History
VCH Vertical Clitoral Hood (piercing)
VCH Volunteer Clearing House (University of Colorado)
VCH Vliegclub Hoogeveen
VCH Virtual Channel Handler
, New York, USA.

[32] IUPAC IUPAC: see International Union of Pure and Applied Chemistry.  Physical Organic Chemistry Physical organic chemistry is the study of the interrelationships between structure and reactivity in organic molecules.[1] It can be seen as the study of organic chemistry using tools of physical chemistry such as chemical equilibrium, chemical kinetics,  Glossary A term used by Microsoft Word and adopted by other word processors for the list of shorthand, keyboard macros created by a particular user. See glossaries in this publication and The Computer Glossary. , 1999, IUPAC, Pure Appl. Chem., 71(10), 1919-1981.

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[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 Microsoft's primary desktop applications for Windows and Mac. Depending on the package, it includes some combination of Word, Excel, PowerPoint, Access and Outlook along with various Internet and other utilities.  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 Islamic Azad University (Persian: دانشگاه آزاد اسلامی , Dāneshgāh-e Āzād-e Eslāmi) is a private chain of universities in Iran. , P. O. Box 38135-567, Arak, Iran Coordinates:

Arak, (in Persian: اراک) previously known as Soltan-abad, is the center of Markazi province, Iran.
. E-mail: avat_1@yahoo.co.uk

* Sabbatical sab·bat·i·cal   also sab·bat·ic
1. Relating to a sabbatical year.

2. Sabbatical also Sabbatic Relating or appropriate to the Sabbath as the day of rest.

n.
A sabbatical year.
Address: Chemistry Building, School of Molecular and Microbial microbial

pertaining to or emanating from a microbe.

microbial digestion
the breakdown of organic material, especially feedstuffs, by microbial organisms.
Science, The University of Queensland The University of Queensland (UQ) is the longest-established university in the state of Queensland, Australia, a member of Australia's Group of Eight, and the Sandstone Universities. It is also a founding member of the international Universitas 21 organisation. , 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

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
```
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