CHARACTERISTICS GRAPH OF SOME HYDROARBONS
1
MOLECULAR (GRAPH) CHARACTERISTICS OF SOME HYDROCARBONS THROUGH GRAPH THEORY
B. K. Mishra Centre of Studies in Surface Science and Technology, Department of Chemistry, Sambalpur University, Jyoti Vihar 768 019, India
ABSTRACT
An organic molecule can be represented by a graph, which can be converted to several matrices by using various graph characteristics. Connectivity of atoms through bonds leads to adjacency and distance matrices. The polynomials, generated from these matrices may be treated as the signature of those molecules. The eigen values of these polynomials are also treated as molecular descriptors and have been used in quantitative structure property/activity relationships. Similarly the relationship of polynomials of a molecule with those of the synthon components of the molecule leads to in silico synthesis.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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MOLECULAR
CHARACTERISTICS
OF
SOME
HYDROCARBONS THROUGH GRAPH THEORY 1. GRAPH THEORY AND CHEMISTRY
1.1. Introduction
To bring the power of mathematics to bear on real-world problems, the problem should be first modeled mathematically. Graphs, representatives of mathematics, are remarkable versatile tools for modeling. A graph, G (V,E), can be defined as a mathematical structure consisting of a vertex set(V) and an edge set (E). Each edge has a set of one or two vertices (also termed as nodes) associated to it. Graph Theory (GT) is largely applied to the characterization of chemical structures, as well as to qualitative and quantitative structure-property (QSPR) and structure-activity (QSAR) relations by means of certain numerical characteristics, the topological indices. Chemical graph theory is a branch of graph theory that is concerned with analyses of all consequences of connectivity in a chemical graph. Chemical graph serves as a convenient model for any real or abstracted chemical system1,2. It can represent different chemical objects as molecules, reactions, crystals, polymers, clusters etc. The common feature of chemical systems is the presence of sites and connections between them. Sites may be atoms, electrons, molecules, molecular fragments, groups of atoms, intermediates, orbitals etc. The connections between sites may represent bonds, bonded and non-bonded interactions, elementary reaction steps, rearrangements, Van der Waals force etc. Chemical systems may be depicted by chemical graphs using a simple conversion rule: Site vertex Connection edge
Chemical graph theory (CGT) appears to be one of the most misunderstood areas of theoretical chemistry. Randi3 tried to outline briefly causes for misunderstanding and
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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suggested remedies, including the test on the knowledge of GT and CGT. CGT should be viewed not only as equal to other branches of theoretical chemistry but also as complementary and necessary for better understanding of the nature of the chemical structure. The language of graph theory is different from that of chemistry. Therefore, a graph theoretical terminology4,5,6,7 (table 1) is proposed for standard use in chemistry and for the corresponding chemical terms. Table-1: Mapping of graph theoretical and chemical terms.
GRAPH THEORETICAL TERMS Chemical (molecular) graph Vertex Weighted vertex
CHEMICAL TERMS Structural formula Atom Atom of a specified element (Mostly other than carbon)
Edge (line) Weighted edge
Chemical bond Chemical bond between the specified Elements
Degree of a vertex Tree graph Chain Cycle Bipartite (bichromatic) graph Nonbipartite graph 1-Factor (Kekule graph) Adjacency matrix (A) Characteristic polynomial Eigenvalue of A Eigen factor of A Positive eigenvalue Zero eigenvalue Negative eigenvalue
Valency of an atom Acyclic structure Linear alkane or polyene Cycloalkane or annulene Alternant chemical structure Nonalternant chemical structure Kekule structure Huckel(topological) matrix Secular polynomial Eigenvalue of Huckel matrix Huckel(topological) molecular orbital Bonding energy level Nonbonding energy level Antibonding energy level
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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Graph-Spectral Theory Graph spectrum
Huckel theory Huckel polynomial equation
Molecular graphs are a special type of chemical graphs, which represent the constitution of molecules1,8,9. They are also called constitutional graphs.10,11 When the constitutional graph of a molecule is represented in a two-dimensional basis it is called structural graph. Pogliani12 introduced the complete graphs for the inner core electrons. Konstantinova and coworkers13 reported on the application of information theory to the problem of characterizing molecular structures. The information indices based on the distance in a graph are considered with respect to their correlating ability and discriminating power. Topological indices are structural invariants based on modeling of chemical structures by molecular graph. Predih and Predih14 reported that the atom contribution approach, the bond contribution approach, the contribution of terminal or interior atoms or bonds approaches do not seem to be viable to perform the structural interpretation of topological indices and physiochemical properties. On the other hand, the approach of structural interpretation using the structural features like the size of the molecule, the number of branches, the position of branches, the separation between them, the type of branches, and the type of branched structure is generally applicable. Perdih15 derived branching indices of the BIA groups. Gutman and coworkers16 derived relations between topological indices of large chemical trees. Turker17 starting with the concept of T (A) graphs for alternant hydrocarbons defined a novel topological index, which differentiates isomeric as well as isospectral molecules very well. Torrens18 derived a new chemical index inspired by biological plastic evolution. 1.2. APPLICATIONS OF CHEMICAL GRAPH THEORY The expensive and time consuming process of drug lead discovery is significantly accelerated by efficiently screening molecular libraries with a high structural diversity and selecting subsets of molecules according to their similarity towards specific collections of active compounds. To characterize the molecular similarity/diversity or to quantify the drug-like character of compounds the process of screening virtual and synthetic combinatorial libraries uses various classes of structural descriptors, such as
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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structure keys, finger prints, graph invariants and various topological indices computed from atomic connectivities or graph distances. An efficient algorithm for the computation of several distance based topological indices of a molecular graph from the distance invariants of its sub-graphs is reported by Klein and Ivancius.19 The procedure utilizes vertex- and edge- weighted molecular graphs representing organic compounds containing heteroatoms and multiple bonds. These equations offer an effective way to construct weighted molecular graphs and consequently to compute the Wiener index, even/odd wiener index and resistance distance index. The proposed algorithms are especially efficient in computing distance-based structural descriptors in combinatorial libraries without actually generating the compounds, because any distance-based indices of the building blocks are needed to generate the topological indices of any compounds assembled from the building blocks. Wiener-type indices have been successfully used in quantitative structure activity/property studies20, 21, 22, 23. The symmetry group of a nonrigid molecule is related to that of the transition structure that is related to the rearrangement process, which contributes to the nonrigidity of the molecular system. The resulting permutation/ rotation/ reflection groups for nonrigid molecules can be much larger in order, than the usual LonguetHuggins24 permutation/inversion group. By the group theoretical approach Bytautas and Klein25 defined the symmetry group for nonrigid molecules. A qualitative resonance-theoretic26 view is presented for the description of a variety of conjugated -network species identified with sub-graphs of a graphite network. Within the framework of this resonance theory, simple rules are described to provide qualitative information on ground state spin multiplicities; on patterns of groundstate spin density; and on exchange splitting to low lying spin-flipped excited states. Beyond ordinary benzenoid molecules, illustrative applications are noted to be a diversity of extended species, including differently structured edges on semi-infinite graphite; corner structures (where edges along different directions meet); conjugated polymer-strip ends; and local defect vacancy structures in extended graphite. The varieties of simple resonance-theoretic predictions are compared against a semi-empirical unrestricted Hartee-Fock view of some quantitative tight binding molecular-orbital computations.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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Agreement in predictions from the resonance- and band-theoretic viewpoint is taken to engender reliability of the predictions. A traditional organic chemical resonance-theoretic view is thence conveniently reformulated and brought to bear on several extended nanostructured systems to reveal systematic patterns of -electronic behaviour. The effects of different types of boundaries on graphite fragments are considered as they influence the -electrons. From a simple resonance theoretic argument there are proposed simple structural conditions governing the occurrence of unpaired -electron density near the edges.27 Predictions based on these rules are made for a variety of edge structures. Further, the novel resonance theoretic argument and predictions are strengthened through more elaborate considerations of both the valence bond and molecular orbital theoretic nature, especially for translationally symmetric polymer strips with various types of edges. 1.3. GRAPH THEORETICAL DESCRIPTORS IN QSAR/QSPR STUDIES Molecular graph is an important tool in drugs designing. Descriptors can be engendered from molecular graphs through construction of matrices and graph enumeration. Estrada and coworkers28 made a review on the use of topological indices in drug design and discovery. Natarajan and Nirdosh29 worked on the application of topological indices to QSAR modeling and selection of mineral collectors. Duchowicz30 and coworkers modeled the free-energy of hydrocarbons on the basis of topological indices defined from the distance and detour matrices with in the realm of the QSPR theory. Sharma et.al.31 calculated the excess isentropic compressibility, KsE values employing density values of the binary and ternary mixtures and graph theory. Basak et al.32 employ topostructural (TS) and topochemical (TC) indices, geometrical descriptors and ab initio quantum chemical indices either alone or hierarchically in the development of QSAR models of the aryl hydrocarbon (Ah) receptor binding potency of a set of 34 dibenzofurans. Gupta et.al.33 investigate the eccentric-adjacency topochemical indices to estimate anti-HIV activity of 107 derivatives of 6-(phenylthio) thymines. Agrawal and coworkers34 used the distance based topological indices in modeling antihypertensive activity of 2-aryl-imino-imidazolidines. Khadikar and coworkers35 use equalized
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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electronegativity (eq) in modeling toxicity of nitrobenzene. Results show that more reliable models can be obtained when eq is combined with topological indices. Mishra and coworkers have used graph theoretic parameters for correlating various physico chemical parameters of alkanes36-42 ethers43,44 and some solvents45; critical micelle concentration of some nonionic surfactants46; and mutagenic activities of amino acids47. 1.4. MATRIX REPRESENTATION OF STRUCTURAL GRAPHS For the purpose of reflecting molecular topology and correlating structure and properties quantitatively, graphs are converted in to a mathematical expression, which may be a matrix, a polynomial, a sequence of numbers or a numerical index. The first representation of a molecule using a matrix is due to Huckel matrix, which is used to derive mathematical expressions of an organic molecule. Some important and widely used matrix representations for characterizing a graph are discussed below.
1.4.1. Adjacency matrix
The vertex-adjacency matrix A (G) of a labeled connected graph G with N vertices is a square NN symmetric matrix, which contains information about the internal connectivity of vertices in G. It is defined as, (A)ij= 1, if vertices Vi and Vj are adjacent 0, otherwise (A)ii = 0 (1) (2)
Adjacency matrix finds wide applications in the fields of chemistry and physics48-52 Example of adjacency matrix:
1 2 4 3
Hydrogen depleted structural graph of methyl cyclopropane
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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The vertex-adjacency matrix of G is
0100 A(G)= 1011 0101 0110 The adjacency matrix is symmetrical about the principal diagonal. Therefore the transpose of adjacency matrix A leaves the adjacency matrix unchanged, i.e. AT (G) = A (G) The edge-adjacency matrix (EA(G)) of a graph G is defined as (EA )ij = 1, if edges ei and ej are adjacent 0, otherwise (EA )ii= 0 Example of edge adjacency matrix For the above graph G, the edge-adjacency matrix is 0101 EA(G) = 1011 0101 1110 The vertex- and edge-weighted graph53-60, GVEW, is a graph, of which one or more vertices and edges are distinguished in some way from the rest of vertices and edges. For adjacency matrix of GVEW the equation (1) and (2) should be modified61-63 as (A )ij =1, if vertices vi and vj are adjacent and if edge (vi, vj) is K-weighted =0, otherwise (Aij)=h, if there is a loop of the weight h at vertex (i) in GVEW (6) (4) (5) (3)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
9 (7)
= 0, otherwise
Example
4 5 k k 1 3 2
Hydrogen depleted structural graph of tetrahydropyrrole The adjacency matrix of the above graph is hk00k k 0100 A(GVEW)= 0 1 0 1 0 00101 k0010 The Mobius systems are defined as cyclic arrays of orbitals with one or more generally, with an odd number of phase delocalization resulting from negative overlap between the adjacent 2PZ- orbitals of different signs64. Mobius system can be depicted by Mobius graphs65-67 (GMO) which is defined as (V (GMO), E+(GMO), E- (GMO)) where, V (GMO)=vertex set E+(GMO)=E+ Edge-subsets E-(GMO)=EThe adjacency matrix for Mobius graph is defined as, 1, if vertices v and v are adjacent (A)ij= -1, if vertices v and v are adjacent and edge (vi,vj) is 1 weighted 0, otherwise
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
10 (8)
(A)ii= Example
0
1
3 -1 4
2
Hydrogen depleted graph of cyclobutane The vertex-adjacency matrix of the graph GMO is 0 A (G ) = 1 1 1 0 0 0 0 1 1 1 0
1 0
0 1
If a bipartite graph is labeled that 1,2-----, s are starred and s+1,s+2, ------, s+u unstarred vertices then (A)=0 for 1 I, j s, s+1 I, j s+u. Example
* 2 5 * 4 3 * 6 1
Hydrogen depleted graph of 1,2 dimethyl cyclobutane The adjacency matrix of the above graph is,
000100 000110 A(GMO) = 000111 111000 011000 001000
A directed graph is called digraph. The adjacency matrix of a digraph AD(G) is defined as
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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AD(G)= the number of arcs from Vi to Vj if ViVj and the number of self-loop at Vi if Vi= Vj (9)
The digraphs can find wide applications in the vectorial electron flow in a system or in representing chemical reactions.
1.4.2 Distance matrix
The distance matrix D = D (G) of a labeled connected graph G is a real symmetry NN matrix whose elements, dij are defined as7, 68,69,70 dij = l if ij = 0 if i=j (10)
Where, dij = length of the shortest path i.e. minimum number of between n vertices Vi and Vj. Example
1 3 2 4 5
Hydrogen depleted graph of cyclopropyl cyclopropane The distance matrix D (G) of this graph is 011233 101122 110233 212011 323101 323110 The distance matrix of a labeled edge-weighted graph GEW is a real symmetric NN matrix whose elements dij are defined as71 dij = wij, = 0, if ij if i=j (11)
D (G)=
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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Where, wij = minimum sum of weights of edges along the path between Vi and Vj which not necessarily the shortest possible. Example
1 4 5
2 3
Hydrogen depleted structural graph of tricyclo [2,1,0,03,5] pentane 01322 10231 D (GEW)= 3 2 0 2 1 23203 21130 The distance matrix D (GVEW) of a labeled vertex and edge-weighted graph GVEW is a real-symmetric NN matrix whose element, dij, is defined as72-74 dij = wii if ij = wij if i=j Where wii=weighted of a vertex V wij=minimum sum of the edge-weights Kij along the path between the vertices Vi and Vj which is not necessarily the shortest possible. 1.5. GRAPH MECHANICAL PARAMETERS The intrinsic characteristics of a molecule can be determined from the quantum mechanical parameters obtained by mathematical operations on quantitative (12)
characteristics of constituent atoms. The graphical representation of a molecule delineates connectivity among the constituent atoms, which also provides information on their interactions. Similar quantum mechanical operations can be applied for graph theoretical interactions and various parameters such as, energy, bond order, electron density, graph angle etc. can be obtained.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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1.6. Isomer Enumeration: Isomers are chemical compounds with an identical molecular formula which display at least some differing physical or chemical properties and which are stable for periods of time that are long in comparison with those during which measurements of their properties are made. Compounds with isomers are called unimers75. Traditionally isomers have been classified as either structural isomers or stereoisomers76. Structural isomers or constitutional isomers differ in there structures i.e. in the manner of bonding the atoms in the molecules77. Stereoisomers have identical structures but they differ in their configuration or conformation i.e. in the special architecture of the molecule. The enumeration of isomeric structures is one of the oldest uses of the graph theory in chemistry. There are a number of methods available for the isomer enumeration. 1.6.1 The Cayley generating functions: Cayley78 was first to attempt to enumerate the isomeric alkanes CH and alkyl radicals CNHN+1. He represented the carbon skeletons of alkanes and alkyl radicals by rooted trees in which the maximum vertex valency is four. These trees are also called the Caley trees79. Example The graph theoretical representations of isomeric pentanes C5H12 by means of trees are
and that of pentyl radicals C5H9 are ;
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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1.6.2.Enumeration of Trees: Cayley80 first used the name tree in 1857,although Kirchhoff81 first utilized the concept in his fundamental work of electrical networks in 1847.Cayley developed a generating function for enumeration of rooted trees,
(1 X) A o (1 X 2 ) A1 (1 X 3 ) A 2 (1 X 4 ) A3 (1 X N ) A N 1 = A 0 + A1X + A 2 X 2 + A 3 X 3 + + A N X N
Where X=variable N=number of vertices in the rooted tree AN=number of rooted trees in a given N A0=1 by definition (13)
Ten years later, Jordan82 discovered the existence of the center and the bicenter of the tree. Every tree has a center or a bicenter, but not both. Thus, tree with a center are called centric trees and while those with a bicenter are called bicentric trees. Example
(bicentric tree T1)
(centric tree T2)
Cayley made use of Jordan and enumerated the centric and bicentric tree. The sum of centric and bicentric trees83 produced the total number of isomeric trees for a given N.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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Later Caley84 improved his method by using the generating functions and already known AN numbers for the enumeration of rooted trees. Thus, the trees TN can be counted by means of the formulae: T1=1 T2=(1/2) A1 (A1+1) T3=the coefficient at X2 in (1-X)-A1 T4=(1/2) A2 (A2+1)+the coefficient at X3 in (1-X)-A1 T5=the coefficient at X4 in (1-X)-A1 (1-X2)-A2 And so on. Cayleys counting formulae have many errors. Half a century later Otter derived an elegant formula for counting trees in terms of rooted trees, T (X)=A (X)-(1/2)[A2 (X)-A (X2)] Where A (X) is the counting series for the rooted trees. The enumeration of trees by means of the Otter counting formula is carried out as: A (X)=X+X2+2X3+4X4+9X5+20X6+48X7+--------A2 (X)=X2+2X3+5X4+12X5+30X6+74X7+---------A (X2)=X2+X4+2X6+4X8+---------------------------(1/2)[A2 (X)-A (X2)]=X3+2X4+6X5+14X6+37X7+--T (X)=X+X2+X3+2X4+3X5+6X6+11X7+---------(20) (21) (22) (23) (24) (19) (14) (15) (16) (17) (18)
The coefficients in the equation 12 represent the counts of trees with a given number of vertices. 1.5.8.1.Methods for Enumeration of Alkanes: Cayley82 first enumerated acyclic hydrocarbons by the application of mathematical theory of trees. He enumerated alkanes and alkyl radicals up to 13 carbon atoms, but the number of isomers obtained for C12 and C13 alkanes, and C13 alkyl radicals were incorrect. Almost immediately after the Cayley paper on
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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enumeration of alkanes, a work by Schiff86 appeared in which he correctly counted distinct alkanes, alkenes and alkyl radicals with up to 10 carbon atoms. Schiff also have the same error in counting for C12 as that of Cayley. The errors in computing the number of C12 and C13 alkanes were first corrected by Herrmann five years later. However, none of the above authors and several others was able to produce a reliable method for enumeration of alkanes with large N. The first significant advance after Cayley came in 1931 when Henze and Blair87-91, at the university of Texas and Austin, developed recursion formulae for enumeration of alkanes and related acyclic structures.
1.5.9. Periodic Table for Isomer Enumeration: A formula periodic table92 for the isomer classes of all acyclic hydrocarbons CnH2m has been proposed, with rows and columns respectively specifying numbers of C atoms and half the H atoms. Asymptotic n behaviour of these enumerations is developed, first for fixed degree u n + 1 - m of unsaturation and second for fixed number 2m of H-atoms. The first-set isomer classes increase in size exponentially fast with n, where as with the second set, the isomer-class sizes increase sub-exponentially, as a power of n. Beyond enumeration various properties of the different isomer classes may be surmised to vary in a systematic manner with position in the periodic table. Such systematic property variation is already generally quantitatively understood for the tables alkane diagonal sequence, which is studied in some quantitative detail in three other works93-95. A similar study96 has been made for property variations of fully conjugated acyclic polyenes. The isomer counts reported here has constitute a first step in such a more extensive study of the whole of the current periodic table for acyclic hydrocarbons, with the second step97 concerning distributions of graph invariants with in an isomer class. A third step98 concerns similar distributions for cluster-expansion approximants for selected molecular properties.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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1.5.10. Graph Cyclicity Index: A new graph-theoretic cyclicity index99 C (G) is defined, being motivated in terms of mathematical concepts the theory of electrical networks. This global bond excess conductance index C (G) then is investigated, with a number of theorems as well as some discussion and numerical investigation. It is found that C (G) typically has less degeneracy than the standard cyclomatic number and has some intuitively appealing features. 1.5.13. Graph Energy: Let G be a graph possessing N vertices and let 1,2, ----------,N be its eigenvalues. These eigenvalues will be labeled so that 12 ------------ N If G represents the molecular graph of a conjugated hydrocarbon, then the total electron energy of this hydrocarbon in the HMO approximation is equal to N+E100,101. Here and are the standard HMO-theoretical parameters where as, E is the quantity depending on the eigenvalues of the graph G as E=E (G)+2 i
I =1 N/2
(25)
For the majority of the graphs first N/2 eigenvalues are positive valued and the other half are negative valued. Hence (25) can be written as E (G)=2+i Where + indicates summation over positive eigenvalues. On the other hand for all graphs, (26)
i =1
i = 0
N
(27)
Combining equations 26 and 27 one arrives at
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
N
18
E (G)= |i|
i =1
(28)
The quantity E in equation (28) is traditionally interpreted in terms of total electron energy. Almost all results obtained in the theory of total -electron energy assume the validity of equation (28). Therefore, some scientist call E (G) as a graph invariant defined as the sum of absolute values of all eigenvalues of a general graph G and named it graph energy. The definition of this novel graph invariant was first published in 1978 in a mathematical journal that is not easy to find but later re-stated in the book and elsewhere. Markovic102 approximate the total -electron energy of phenylenes by means of a linear combination of spectral moments of both molecular and line graphs. The idea to use E as a molecular structure descriptor, for predicting physicochemical properties of saturated organic molecules, seems to be first expressed by Rakshit et.al103,
104
. Some ten years later Randic, Vrako and Novi105 put forward an
identical suggestion. They communicated a limited number of correlations that hinted towards the possible applicability of E in QSPR/QSAR studies. The total -electron energy of an alternant hydrocarbon described by graph G (N; e) is given by E= 2 ne cos Where n=N/2 =(N-1)/2, for odd alternant hydrocarbons. [N=number of vertices (atoms)] e =number of edges (bonds) and =angle of total -electron energy. is found to be very much useful in studying and encoding the fine topologies of structural isomers of alternant hydrocarbons. In cases of isospectral structural isomers, their values have to be same as according to the above equation (29). Then, obviously their fine topologies have to be encoded not only by alone but some other topological variants as well. One of them is azimuthal angle106 (). An angle could possibly exist (29)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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for any structural isomer of nonisospectral too but in the case of isospectral structural isomers its role is great108. Now, in a three-dimensional linear space vectors can be defined as C(0,0,n) and Di(xi,yi,z) such that norm Di=e, tan =r/z Where, r =(xi2+yi2). Note that (C, Di)= (ne)cosnz. The above treatment can be generalized including the odd alternant hydrocarbons because these systems possess non-bonding molecular orbital having zero energy. The set of vectors {Di}(generatrix) constitute a vector field (conical surface) in three dimensional Euclideanspace109, 110, which has the form of an upside down cone shown in the Fig-1 whose apex is on the origin of the Cartesian height is z.
Z Di
coordinate system (x, y, z), and its
yi r xi X
Y
Fig-1: Vector Field of a Compound The angle between the vector Di and the z-axis represents the angle of graph energy and the angle between r and xi represents the azimuthal angle where i=arctan(yi/xi). Hence each compound is characterized with four variables n, e, and . Thus graph theoretical enumeration led to establish different characteristic features of organic molecules. For the limitation of undertaking very large molecules in quantum mechanical calculation, graph theoretical methods take the stage by using fragmented but similar groups for various eigen-functions. In the present programme small organic molecules containing six carbon atoms have been subjected to graph theoretical manipulation for determining different eigen functions.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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2. CALCULATION OF GRAPH THEORETICAL PARAMETERS 2.1. Representation of molecules in matrix
For calculations all the possible structures of molecules containing up to 6 carbon atoms and having maximum possible number of conjugated double bonds have been considered. The number of possible structural isomers of C2 is one, which is an ethene molecule. The number of possible structural isomers of C3 is 2 where one is cyclic and other is acyclic. For C4 system the number of structural isomers are found to be 6 where 2 are acyclic and 4 are cyclic. For C5 system the number of structural isomers are found to be 17 out of which 3 are acyclic and 14 are cyclic. From all the possible structural isomers of C6 here 12 are considered. All the structures of the molecules are given in table 1. The hydrogen depleted structural graphs of the above graphs of the above molecules are considered for obtaining matrices for further calculations. All the structural graphs of above 38 molecules are given in the table-1.
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Table-1: Structural graph, adjacency matrix, polynomial and roots of some 38 molecules Graph No. 1
H
Molecular structure
H 1 2 H
Structural Graph
1 2
Adjacency matrix x1 1x
Polynomial x2-1
Roots (x) 1, -1
H
H
x
3 H H H
1 2 3
10
x3-2x
2
H 1 H 2
1x1 01x x 11
0, 1.41421
H
H
1
3
H
1 3 2 H
3
2
1x1 11x
x100 1x11 01x 0
x3-3x+2
-2,1,1
H
H 1 H 2 H 4
1 2 3 4
4
H 3 H
x -3x
4
2
0, 0, 1.73205
H
010x x100 1x10 01x1 001x
H
H
H
2 4 3
5
H 1 H 2
3 4 H H
1
x4-3x2+1
0.61803 1.618034
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H 3
H
x101
2 3 1 4
6
H
2 4 1 1 CH 2 H
1 2 4 3
1x1
0
01x1 101x x100 1x11 01x1 011x x111 1x1 0
x4-4x2
2, 0,0
7
4 H
2 3
x4- 4x2+2x+1
-2.17009, 0.31111, 1, 1
H
H
8
1 2
-2.56155, 0, 1, 1.56155 x4-5x2+4x
4 3
1 2
4 3
11x1 101x x111
H
1 4 3
1 2
4 3
9
2
1x11 11x1 111x x1000
x4-6x2+8x+3
-3, 1, 1, 1
1
CH 2 2 5 CH 2
1 2 3 5 4
10
1x111 01x 00 0 010x
HC 2
x5-4x3
-2, 0, 0, 0, 2
4 CH 2
0100x
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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H
H 4 3 H 5 CH 3 1 2 3 4 5
x1000 1x1 00 0 01x1 x5-4x3+3x 0, 1, 1.73205
11
1 CH 2
2
001x1 0001x x1000 1.847759 0.765367, 0
H
12
4 CH 1 CH 2 2 5 3 CH
2
2 1
3
4
1x1
00
01x11 001x0 0010x x101 0
5
x5-4x3+2x
3 3
5
H
13
H
4 1 3 2
CH
3
1 4 3
1x1
00
H
1 H H
2
01x10 101x1 0001x x1001
x -5x +2x
5
3
0, 2.13578, 0.662153
H H
1 2 3 5 4
14
H
2 3 4
5
1x1
00
H
01x10 001x1 1001x
x5-5x3+5x+2
-2, 0.617999, 0.617999, 1.618035, 1.618035
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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H
1
H 5
2 1 5 4
x1001 1x1 01 0 01x1
15
H
2 3
x -6x +2x +4x
5
3
2
4
3
-2.48119, 0.68889, 0, 1.170087, 2
H
001x1 1101x x1001 -2.85577, 0.32164, 0, 1, 1.7741
H
16
1
H 2 3
1 5 4 2 3
1x11
0
5 4
01x11 011x1 1011x x1001 1x111 01x11 011x1 1111x x1 01 0 0 1x11
x5-7x3+4x2+2x
H
1
H 2 3
1 5 4 2 3
17
5 4
x5-8x3+10x2-x-2
-0.35793, 3.3234, 1, 1, 1.68133
H
18
4 1 2 H 3
CH 3 1 H 2 4 3
5
x -6x +4x +2x
5
3
2
01x10 111x1 0001x
0, 2.68554, 0.3349, 1.27133, 1.74912
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
25
x1000
H
19
3 4 2 5 1 CH
4 3 2
5
1x1
1
01
x -6x +4x +3x-2
5
3
2
01x11 001x1 0111x x1000
-2.64119, 0.72374, 0.58922, 1, 1.77571
H
2
H
4 5
3 2
20
4 5 2
1x111
1
3 1 CH
01x11 011x1 0111x x111
5
x5-7x3+8x2-2
-3.08613, 0.42801, 1, 1, 1.51414
3
5
H H
0 0
21
2 1 4 3
1x1
01
x5-7x3+4x3+2x
2 1
H 1 H 4 3 5 2
3 4
1
11x1
-2.85577, 0.32164, 0, 1, 2.17741
101x1 0101x x101 0 x5-7x3+6x2 -3, 0, 0, 1, 2 1x111 01x10 111x1 0101x
22
4 3
5 2
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
26
x1001
H
23
5 4 1 3 2
5 4
H
1x1
1
01
3
2
01x11 001x1 1111x
x5-7x3+6x2+3x-2
2.93543, 0.61803, 0.4626, 1.47283, 1.61803 2.21432, -1, 0.53918, 1, 1.67513
H
1 2 3 4 5
CH
1 2 2 3 4 5
x1000 1x1 00 01x11 001x1 0011x x1000 1x111 x5-5x3+2x2+3x x5-5x3+2x2+4x-2
24
H
H
H
H
25
1 CH 2 2
3 4
H 5 CH
3 1
2
2
4 5
01x11 011x1 0001x x11 00 00 x5-6x3+4x2-5x-4
-2.30278, 0.61803, 0, 1.30278, 1.61803
H
26
H 2 1 3 4 5 H
2 1 3 4 5
1x1
11x11 001x1 0011x
2.56155, -1, 1, 1, 1.56155
H
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
27
x10000 27
H 2 3 CH 3 6 4 CH CH 3 5 3 3 6 4 5 1 CH 2 2 1
1x
1000 x -5x +3x
6 4 2
01x111 001x 00 0010x0 00100x x10000 1x 1001
0, 0, 0.835, 2.07431
28
CH 12 2 6 CH 3
4 CH 3
2
1 2 6
4 3 5
CH 3 5
01x110 001x00 0010x 0 01000x x10000
x6-5x4+4x2
0, 0, 1, 2
5
29
5 CH 4 6 CH 3
3 3
H
6
4
3 2 1
1x1000 01x100 001x11 0001x 0001 0 0x
x -5x +5x
6
4
2
2 H
0, 0, 1.17554, 1.90211
CH 12
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
28
x10000
H H
30
1 CH 2 6 CH 2 3 2 4 5 CH 2
3 1 2 6 4 5
1x
1000 01 0 x6-5x4+5x2-1
01x1
0.51764, 1, 1.93185
001x1
0001x0 00100x x10000 1x
5
H
31
CH 2 1 2
3 4 H 5 H
CH 6 1
2 3
4
1000 x6-5x4+6x2-1
01x100 001x10 0001x1 00100x x10000
0.44504, 1.24698, 1.80194
32
H 5 6 1 H CH 2
H 4 3 2 H
5 6 1
4 3 2
1x1000 01x101 001x1 0 0001x1 00101x
x6-6x4+6x2
0, 0, 1.12603, 2.17533
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
29
x10001 33
H 5 6 H 4 3 2 H 1 H
5 6 1 4 3 2
1x
1000 01 0
01x1
x6-7x4+7x2-1
001x1
1, 0.41421, 2.41421
0001x1 10101x x10000
1 H 6 2 3 5 H 4
1 CH
34
H 5 H
H
2 2 3 4
1x
1001 x6-6x4+8x2-1
6
01x100 001x10 0001x1 01001x x10001
0.37309, 1.32132, 2.02852
35
H 6
1
1
2 3 H
2 3
6
H 4 H
1x1000 01x100 001x1 0 0001x1 10001x
x6-6x4+9x2-4
1, 1, 2
H
5
5
4
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
30
x10001
H
1
36
1 3 5
2
6 3
2
1x
11
00 01 0
01x1
4
x6-8x4+2x3+10x2+2x-1
011x1
H
4
5
0001x1 10101x
2
-2.74108, -0.71029, -0.61803, 0.23136, 1.61803, 2.22001 -3, -1, 2, 2, 0, 0
1 1
x10011 1x 11 00 01x101 011x10 1001x1 10101x x11000 x6-9x4+4x3+12x2
37
6 5 3
2 6 5 4 3 4
H
2 1 3 4 6 5
1 3 4 6
2
38
1x1010 11x100 001x11 0101x1 00011x
x6-8x4+4x3+12x2-8x
-2.73205, 1.41421, 0, 2, 1.414212, 0.73205
H
5
For converting the graph into a matrix here we have considered only the adjacency matrices of the molecules. The adjacency (A) matrix can be defined as, Aij=1, if vertices Vi and Vj are adjacent =x, if vertices Vi=Vj =0, otherwise For example for molecule (I)
H 3 HC 3 2 1 CH 2
(30)
(I)
2 3 1
the structural graph is
and hence its adjacency matrix by definition (1) can be
x
represented as
1
0
1x1 01x
All the adjacency matrices of the considered 38 molecules are given in the above table-1.
2.2. Determination of Characteristic Polynomial: The polynomials containing x as variable are derived from the adjacency matrices of the molecules by the determinant evaluation method. All the polynomials for the molecules are given in table-1. Previously some scientists111-114 had derived recurrence relation for linear polyenens LN with N vertices whose characteristic polynomials are symbolized by P(LN,x). The polynomials up to L20 are given below: P (L0; x)=1 P (L1; x)=x P (L2; x )=x2-1 P (L3; x)=x3-2x P (L4; x)=x4-3x2+1
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
32
P (L5; x)= x5-4x3+3x P (L6; x)=x6 5x4 +6x2-1 P (L7; x)= x7-6x5+10x3-4x P (L8; x)= x8-7x6 +15x4-10x2+1 P (L9; x)= x9-8x7+21x5-20x3+5x P (L10; x)= x10-9x8+28x6 -35x4+15x2-1 P (L11; x)= x11-10x9+36x7-56x5+35x3-6x P (L12; x)= x12-11x10+45x8-84x6 +70x4-21x2+1 P (L13; x)= x13-12x11+55x9-120x7+126x5-56x3+7x P (L14; x)= x14-13x12+66x10-165x8+210x6 -126x4+28x2-1 P (L15; x)= x15-14x13+78x11-220x9+330x7-252x5+84x3-8x P (L16; x)= x16-15x14+91x12-286x10+495x8-462x6 +210x4-36x2+1 P (L17; x)= x17-16x15+105x13-364x11+715x9-792x7+462x5-120x3+9x P (L18; x)= x18-17x16+120x14-445x12+1001x10-1287x8+924x6 -330x4+45x2-1 P (L19; x)= x19-18x17+136x15-560x13+1365x11-2002x9+1716x7-792x5+165x3-10x P (L20; x)= x20-19x18+153x16-680x14+1820x12-3003x10+3003x8-1716x6 +495x4-55x2+1 A recurrence relation for computing P (LN; x) has also been suggested: P (LN; x)= x P (LN-1; x)- P(LN-2; x) For example: The polynomial for linear polyene L5 can be obtained by equation (31). P (L5; x)= xP (L4; x)- P (L3; x) Where, P (L4; x)=x4-3x2+1 P (L3; x)=x3-2x Putting these values in equation (32) P (L5; x)= x(x4-3x2+1)-(x3-2x) = x5-4x3+3x Cyclic CN may be used to represent the carbon skeletons for cycloalkanes and [N] annulenes. To compute P (CN; x) a recurrence relation has been obtained P (CN; x)= P (LN; x)-P (LN-2; x)-2 (33) (32) (31)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
33
Thus the polynomial for the cyclobutane, C4, can be evaluated by equation (33). P (C4; x)=P (L4; x)-P (L2; x)-2 Where, P (L4; x)=x4-3x2+1, P (L2; x)=x2-1 Putting the above values in the equation (34) P (C4; x)= (x4-3x2+1) (x2-1) 2 =x4-4x2 A recurrence formula for computing the polynomial of the side chain-extended graph of a cyclic system has been derived. The side chain extension is mostly due to a single bond extension at any vertex of the graph. An isolated single bond is due to a connection of two atoms (vertices). The polynomial of a side chain-extended graph of a cyclic system (GN) can be represented as P (GN; x). The recurrence relation is P(G N ; x )( x 2 1) -axr +dGN x (34)
P(GN; x)=
(35)
Where, a is the degree of the carbon where the new bond is added when the number of carbon atoms is >4 otherwise zero. r is the power of last even term when the number of carbon atoms is even and last odd term when the number of carbon atoms is odd. dGN is the determinant of the GN graph. In the equation (35) the constants and the negative power of x of the first term are neglected. In the chain-extension process initially a single carbon atom (x) is added to the ring by a naked bond. A naked bond can be represented as P (G2; x)/x2, where x represents individual atom. For the addition of an atom (x) to a cyclic graph P (GN; x) the addition should be P (GN; x) P (G2; x)/x2 x Where, P (GN; x) represents the cycle, x represents the atom and P (G2; x)/x2 represents the naked bond.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
34
For example:
The polynomial of P (G5;x)=
can be derived from
by equation (34). (36)
P(G 4 ; x ) ( x 2 1) -2x+dGN x
Where the constant term and the terms having negative power of x are neglected from the first term of the equation (36).
P (G5; x)=
( x 4 5x 2 + 4 x ) ( x 2 1) -2x+dG5 x (37)
=(x5-6x3-4x2+5x-4)-2x+dG5 Neglecting the constant terms from (37) P (G5; x) = (x5-6x3+4x2+5x)-2x+dG5 =x5-6x3+4x2+3x+dG5
01000 10101 Again, dG5= 0 1 0 1 1 =(-2) 00101 01110
P (G5; x)=x5-6x3+4x2+3x-2
The antithetic analysis of a cyclic graph reveals that a cycle can be generated from prestruct with a single bond disconnection i.e. from a single species or may be with two bond disconnections i.e. from two isolated species. An addition of a naked bond to two atoms for the former case can lead to a cyclic product. For the later class of prestruct, the isolated species may be an atom (vertex) or may be a C2 or more ordered graph with two
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
35
terminal atoms (vertices) to be bonded. In either case two bonds are to be connected with no addition of atoms. To determine the polynomial a recurrence formula can be proposed. P (GN; x)=P (GN-2; x) P (G2; x)-2xN-2+axr+dGN Where P (GN-2; x) itself represents a complete graph P (G2; x) is the polynomial of a C2 graph. a is zero for number of carbon atoms 4 or is equal to 3 and 1 alternatively when number of carbon atoms > 4. r is the power of the last even term when number of carbon atoms is even and of the last odd term when the number of carbon atoms is odd. dGN is determinant of the GN graph. In the above equation the constant term of the first term is neglected.
(38)
For example:
P (G3; x)=P (G1; x)P (G2; x)-2x+0+dG3 = x (x2-1)-2x+2= x3-3x+2
P (G4; x)=P (G2; x)P (G2; x)-2x2+0+dG4 = (x2-1) (x2-1)-2x2 = x4-4x2+1 Neglecting the constant term from the first term P (G4; x)=x4-4x2
P (G5; x)=P (G3; x) P (G2; x)-2x3+3x+dG5 =(x3-2x) (x2-1)-2x3+3x+2=x5-5x3+5x+2
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
36
P (G5; x)=P (G3; x)P (G2; x)-2x3+x+dG5 =(x3-3x+2) (x2-1)-2x3+x+0
Neglecting the constant term in the first term of the above equation P (G5; x)=x5-6x3+2x2+4x
P (G6; x)=P (G4;x) P (G2; x)-2x4+3x2+dG6 =(x4-4x2) (x2-1) 2x4+3x2-1 =x6-7x4+7x2-1 The above procedure of deriving polynomial may help in synthesizing molecules in silico. Patra et al115 have reported the building up of a reaction scheme through adjacency matrix to explain the generation of some interstellar molecules with various reference frame for the reactants and reagents. They have proposed probable mechanisms during the generation of some interstellar molecules.
2.3. Determination of the Eigenvalues: The roots or eigenvalues of variable x are calculated. All the roots of the considered 38 molecules are given in the table-1. The sum over of the roots of each graph is found to be zero indicating the validity of the method. For the bipartite or alternant graph we get a pair of roots where as for nonbipartite or nonalternant graph we do not get pairing in the eigenvalues.
For example; For
* *
roots are x=0, 1.41421
And for
*
*
roots are x=-2, 1, 1.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
37
From the above-calculated roots of variable x, the eigenvalues with most negative value is regarded as ground state orbital and the wave function is designated as 1. Then 2,3are notified in increasing order. Each root of a molecule y number of coefficients are calculated where y is same as the number of carbon atoms in a molecule. To calculate the coefficient values first the cofactor polynomials are derived from its adjacency matrix designated as C1, C2, CN. All the cofactor polynomials are then divided by the C1 polynomial and there after putting the corresponding x value we get C1/C1, C2/ C2, -----, CN/C1. Then from these value we get (C1/C1)2, (C2/C1)2(CN/C1)2,(Ci/C1)2 and , ((Ci/C1)2)1/2 .Then the above calculated values of C1/C1,C2/ C2,-----,CN/C1 are divided by ((Ci/C1)2)1/2 to get the values of coefficients C1,C2------------CN. From the values of coefficients and roots various parameters such as bond order, electron density, free valence index, energy, graph angle etc. are calculated. 2.4. Determination of Bond Order: The bond order of a molecule is calculated from its coefficient values by the formula. Pab=n1(Ca)1(Cb)1+n2(Ca)2(Cb)2+-------------+nn(Ca)n(Cb)n (39)
When n1, n2, --------, nn are number of electrons in 1st, 2nd, -----nth shell respectively. The values of bond orders calculated from the above equation for all the considered molecules are given in the table-2. Table-2: Graph theoretical bond order values of some small organic molecules. Graph No. 1
2
Molecular structure
1
2
Number Bond Orders of pi electrons 2 P12=1.0003 2 2 P12=0.7069, P23=0.7073 P12=P23=P13=0.666551
2 3
3
1 3 2
1
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1
38
4
3
2
2 4
4 3
2
P12=P23=P24=0.577349
5
1
4
P12=0.886774,P23=0.453931, P34=0.883404
2
3 4
6
1
4 4
P12=P23=P34=P14=0.5 P12=0.758295,P23=P24=0.458673, P34=0.817554 P12=P23=P34=P14=P13=0.498319 P12=P23=P34=P14=P13=P24=1
1
7
4
1
2 3
4 3 4
8 9
2 1 2
1
4 4
3
2
10
3 4
4
5
P12=P23=P24=P25=0.5
2
4
11
1 3
4 1 2 5
4
5
P12=P45=0.788675, P23=P34=0.577345
12
3
4
5
P12=0.560457, P23=0.651527, P35=P34=0.328016 P12=P23=0.610131, P34=P14=0.357408, P45=0.862856 P12=P15=0.644891, P23=P45=0.199959, P34=0.928354
13
1 2
4
4
3
1
14
2 3
1
5
4
4
15
2 3
5 4
4
P12=P15=0.701555, P23=P45=0.551942, P34=0.977456, P25=0.538822
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1
39
16
5 4
1
2
4
3
P12=P15=0.506149, P23=P24=P35=P45=0.345784, P34=0.781979 P12=P15=0.313076, P34=0.384965, P23=P45=P24=P35=0.447831
17
5 4
2 3
4 3 5
2
18
1 2
2
P12=0.439004, P23=0.431429, P34=0.479917, P45=0.252726, P24=0.610133, P14=0.479914 P12=0.793929, P23=P25=0.343464, P34=P45=0.58821, P35=0.631675 P12=0.472345, P23=P24=P25=0.140917, P34=P45=P35=0.129742 P12=P34=P23=P14=0.346219, P13=0.782069, P25=P45=0.505228
3
19
5
4
4
2
3
1
20
5
2
2 1
5
21
2 1
3 4
1
4
22
4 3 5 2
6
P12=P23=P34=P45=P14=P25=-0.0000004, P24=0.499997
5
23
4 1 3 2
4
1
P12=P34=0.776694, P23=0.165328, P45=P15=0.3919299, P35=P25=0.525445
24
4
2 3 5
3
4
P12=0.816387, P23=0.41352, P34=P35=0.600266, P45=0.694325
25
1
2
4 5
4
P12=P45=0.724572, P23=0.554702, P34=0.556998, P24=0.362278 P12=P45=0.81063, P13=P23=P34=P35=0.485071
2
1
26
4
3 5
4
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
40
27
6
2 3
1
2
4
P12=0.174182, P23=0.57531, P34=P35=P36=0.441607
5
1
4 2 3 5
5 2
28
6
4
P12=P26=0.666666, P23=0.333333, P34=P35=0
29
6 4 3
4
1
P12=0.760822, P23=P45=P46=0.615539, P34=0.470218 P12=0.909231, P45=0.908147, P23=0.407488, P34=0.408376, P36=0.816244 P12=P56=0.87113, P23=P45=0.483432, P34=0.784851 P12=0.797554, P34=P45=P56=0.550225, P34=P36=0.446563
2
4 3 5 6
3 6 4
5 4 3 2 4 3 2 1 2
30
1
6
31
1
2
5
6
32
6 1 5
6
33
6 1
6
P12=P45=0.499977, P23=P56=P34=0.7071117, P16=0.707076, P36=0.000007
34
6 5
1
3 4
6
P12=0.021745, P23=P26=0.651483, P34=P56=0.275564, P45=1.224986
35
6 5 4
2 3
P12=P23=P34=P45=P56=P16=0.333333 6
1
2 6 3 4
36 6
5
P12=0.462983, P23=P34=0.611843, P45=0.463026, P56=0.682228, P36=0.102032, P24=0.453053, P61=0.682369
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
41
1
2 6 3 4
37 6
5
P12=0.666664, P23=P24=0.33333, P34=P56=0.833341, P45=P36=0.166669, P16=P15=0.33555 P12=P13=0.142229, P23=1.14434, P34=P25=0.144335, P45=0.644333, P45=P56=0.642226
38
1 3 4 6
2
6
5
For convenience, during the calculation of bond order the electrons for double bonds are only considered. The numbers of electrons considered in each case are given in table 2. Hence the bond orders are due to pi-electron system only and are found to be 1 unit less than the value obtained by normal quantum chemical method. For example, the bond orders for butadiene116 are reported to be 1.894 and 1.447, whereas in the present case the values are found to be 0.887 and 0.454. Similarly for 3-methylene penta-1, 4diene, the bond orders obtained from quantum chemical calculation are found to be 1.930, 1.859 and 1.363 whereas the corresponding values obtained from graph theoretical technique are0.909, 0.816 and 0.408 respectively.117 The values obtained in the present case are referred to as graph theoretical bond orders (GTBO). The trend in GTBO values depicts the resonance characteristics in a molecule. Molecules with identical resonating structures have same GTBO for similar bonds. When the delocalization is restricted there is a change in the GTBO values for the bonds. As in case of 7, the GTBO values of 1,2 and 1,3 bonds are found to be 0.758 and 0.818 respectively, while the other two bonds have the value 0.459. Similar is the case of for 26. 2.5. Determination of Electron Density: The closed shell electron density of a molecule is calculated from its coefficient values by the formula qa=n1 (Ca)2+n2 (Ca)2+n3 (Ca)2+ ----------- +nn (Ca)2 (40)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
42
When n1, n2, ----nn are number of electrons in 1st, 2nd, ------nth shell respectively. The values of electron densities and there summation for all considered molecules are given in table-3. Table-3: Electron densities of some organic molecules calculated using closed shell model N Graph q1 q4 q2 q3 q5 q6 qi No. q =1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1.0003 1.000264 0.666551 0.333331 0.99495 1.5 1.480864 1.541267 1 0.25 0.666667 1.20548 0.500001 1.191814 0.967445 1.535518 0.188145 0.339348 0.832192 1.45772 0.782074 2.93329 0.969359 0.847846 0.844047 1.0003 0 0.666551 1 1.083691 0.5 0.876748 0.459035 1 1 1 1.041097 1 0.47574 0.538822 0.450264 0.520963 0.548488 0.956843 0.153054 0.449959 0.6 0.718119 1.034595 0.915072 1.000264 0.666551 0.333335 1.083691 1.5 0.817554 1.541267 1 0.25 0.66667 0.767129 0.500001 0.928354 0.977456 0.781979 0.384965 0.339353 0.631675 0.129742 0.78207 0.9333384 0.718121 0.728915 0.481763 0.333335 0.998029 0.5 0.817554 0.459035 1 0.25 1 0.493145 1 0.928354 0.977456 0.781979 0.384965 0.678706 0.749062 0.129742 0.449959 0.6 0.969364 0.694325 0.915068 0.25 0.66667 0.49314 1 0.47574 0.538822 0.450264 0.520963 0.094106 0.63167 0.12974 1.53593 0.933338 0.62504 0.69432 0.84404 2.0006 2.000528 1.999653 2.000001 4.160361 4 3.99272 4.000604 4 2 4.000004 3.999996 4.000002 4.000002 3.994209 4.000004 1.999971 2.000001 3.801447 2 3.999992 5.9999668 3.999999 4.000006 3.996418
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
43 0.81063 0.21289 0.5 0.40001 0.99978 1.00001 1.33333 1.00004 1.22498 0.99999 1.14705 0.83334 0.64433 0.212894 0.5 0.400006 0.999506 1.000008 0.499996 0.999946 0.66288 0.999999 1.13716 0.833338 1.211324 3.999986 2.000002 3.999998 4.000001 6.000004 6.000032 5.999998 6.000019 6.119977 5.999994 6.000006 6.000016 5.999994
26 27 28 29 30 31 32 33 34 35 36 37 38
0.81063 0.08397 0.5 0.599942 0.999748 1.000008 1.333334 0.999983 1.136745 0.999999 1.147195 1.333332 1.211324
0.81063 0.36132 0.999999 1.000029 1.001062 1.000008 1.000007 0.999995 1.2075 0.999999 1.005789 1.333332 1.14434
0.757466 0.91603 0.999999 0.600018 0.999939 1 1.333332 1.00004 0.66288 0.999999 0.556915 0.833338 1.14434
0.81063 0.212894 0.5 1 0.99997 1 0.499996 1.000012 1.224986 0.999999 1.005899 0.833338 0.644333
The open shell electron densities of a molecule are calculated by considering at least one electron at each valence state. Table 4: Electron densities of some small organic molecules using open shell model Graph No. q1 1.000264 0.999732 1.666587 1.666666 0.99459 1.5 2.744032 2.081267 1 1.750002 0.999999 1.205482 q2 1.000264 1.000264 0.666607 1 1.003691 0.5 0.438414 2.081267 1 1 1 1.041097 1.000332 0.666607 0.666669 1.003691 1.5 0.408777 0.418885 1 0.416666 0.999999 0.767128 0.66669 0.998029 0.5 0.408777 0.418885 1 0.416666 1 0.493145 0.416666 0.999999 0.493145 q3 q4 q5 q6
q =1
qi
N
1 2 3 4 5 6 7 8 9 10 11 12
2.000528 3.000328 2.999801 4.000004 4.000001 4 4 5.000304 4 4 5 3.999997
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
44 1 5 5 4.999988 5 3 4.00001 4 2.000004 3.999988 2.999998 3.999982 4 5.004466 3.000008 1.5 8 6
13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
1 1.738614 2.8002 2.418177 2.094072 0.919664 1 1.377309 1 1 0.861804 0.999996 0.999993 0.499992 0.999993 1.5 1.399943 0.999745 0.993275 1.333335 0.999983 1.623182 0.999999 1.000036 0.999999 1.249997
1 0.82441 0.551306 0.581064 0.260482 0.420695 1 0.35937 0.499996 0.499997 0.638188 1.00001 0.99999 0.500004 0.999995 0.999999 1.000029 1.001063 1.103704 1.000007 0.999995 1.227593 0.999999 0.999975 0.999999 0.749999
1 0.80628 0.548588 0.709844 0.192482 0.919692 0.5 0.087775 1 0.500002 0.638188 0.999994 0.999992 1.000008 1.000006 0.999999 1.400019 0.999933 0.965897 1.333333 1.00004 0.830073 0.999999 0.999999 0.500005 0.749999
0.999999 0.80628 0.548588 0.709844 0.192482 0.839347 1 0.087775 0.4999996 0.499997 0.861804 0.5 0.99999 0.500002 0.333336 1.5 0.999999 0.99997 0.965897 0.499996 1.000012 0.744533 0.999999 1.000032 0.500005 0.749999
0.82441 0.551306 0.581064 0.260482 0.900612 0.5 0.087775 0.999996 0.500002 0.999998 0.5 1.004501 0.500002 1.5
0.333336 0.333336 4.000074 0.600006 0.600006
0.999748 0.999506 5.999968 1.103704 0.993275 6.125752 1.333333 0.499996 6 1.000043 0.999946 6.000019 0.744533 0.830073 6.048324 0.999999 0.999999 5.999992 0.999962 0.999995 5.999999 0.500005 0.500005 4.000018 0.749999 1.249997 5.49999
The charge densities on each atom were calculated by using quantum mechanical models on graph theoretical results. The values are found to be in the same trend as in case of bond order. However, the antiaromatic characteristics of the cyclobutadiene can be significantly marked from the charge density values. The values at alternate carbons
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
45
are found to be same and different for adjacent atoms. For 4 and 10 the central carbon has the highest values (=1) and the other carbon atoms share equal values sum totaling to 1. In the open shell method, an equal distribution of electrons is assumed for each carbon atom and thus a change in the vales are obtained in a few cases. A significant change has been observed in 4, where the central carbon retains its value (=1) while one of the atoms has a value one unit more than other two carbon atoms.
2.6.Free Valence Index:
The free valence index of a carbon atom of a molecule is calculated by the formula
Fa= 4.732-[m PCa H + r PCa Cb + s PCa Cc + q PCa Cb + o PCa Cc ]
(41)
Where, m=number of Ca-H bond r=number of Ca-Cb -bond s=number of Ca-Cc -bond q=number of Ca-Cb -bond o=number of Ca-Cc -bond
and PCa H , PCa Cb , PCa Cc , PCa Cb , PCa Cc represent the bond order of corresponding bonds.
The values are given in table-5. Table 5: Free valence index at different atoms of some small organic molecules. Molecular structures
1 2
2 3 1 1
F1 0.729 1.0251 -0.6011
F2 0.729 0.3178 3.0651
F3
F4
F5
F6
1.0297 0.398898
3 2 1
2 4 3
0.8452
0.3913
0.3947
0.8486
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1 2 3
2 1 3 4
46
0.1547
4
-0.00005
1.1547
0.1547
0.732 0.9737
0.732 0.0564
0.732 0.4561
0.732 0.4561
1 2 4
1 2 1 2 1 2 3 5 4
3
4 3 4
0.7354 0.232
0.2763 0.232
0.7353 0.232
0.2763 0.232
3
1.232
4
-1.268
0.232
0.232
0.232
2 1
1 2 4
0.9433
3
5
0.366
0.5773
0.366
-0.0567
5
0.7645
3
0.5117
0.7645
0.1543
-0.1309
1 2 3
1 2 3 1 5 4 1 5 4 2 3 2 3 5 4
5 4
-0.5578
0.8872
0.6037
0.6037
0.8872
-0.6711
-0.0603
0.2026
0.2026
-0.0603
-0.2803
0.5343
0.2585
0.2585
0.5343
0.1059
-0.7898
0.4514
0.4514
-0.7898
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
5
47
1 2
4 3
0.8131
3
0.2514
-0.1794
-1.0907
0.4793
0.9381
5
4 5 2
0.2511
0.1687
0.5556
0.1687
2
3
1
0.2597
1
4
-0.1631
0.3316
1.3316
1.3316
1
2 5
3
1.1715
5
0.5200
0.4244
0.404
1.404
2 1
3 4 1 5 3 2 5
0.2575
0.5343
0.2575
0.5343
-0.2785
4
-0.068
-1.068
-0.068
-1.068
-0.68
4 1 3 2
0.5634
1 2 3 4 5
3 2 4 5
0.2645
0.2645
0.5634
-0.1028
0.9156
0.5021
0.118
0.4374
-0.5626
1.0074
0.0905
-0.3797
0.0882
1.0074
1
2 3 4 2 3 6
1
0.4363
5 1
0.4363
0.4363
0.4363
-1.2083
0.5772
4
1.0829
0.5461
0.6314
0.6314
0.6314
5
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1 2 6
5 2 6 4 3 1
48
4 3 5
1.0653
0.0653
1.3987
1.732
0.732
0.0653
0.9712
0.3556
0.6462
0.6462
0.1165
0.1165
2 1
4 3 5 6
3 6 4
4 3 2 4 3 2 1 2
0.8228
0.4153
0.0999
0.4155
0.8239
0.9158
0.8609
0.3774
0.4637
0.4637
0.3775
0.8608
1
2
5 6 1 5 6 1
5
0.9345
0.3842
0.2886
0.7352
0.6316
0.7352
0...