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On a quantum chemical interpretation of molecular connectivity indices for conjugated hydrocarbons
THEO CHEM Journal of Molecular Structure (Theochem) 342 (1995) 173-179
On a quantum chemical interpretation of molecular connectivity indices for conjugated hydrocarbons Ivan V. Stankevich”>*,
Mariya
I. Skvortsovab,
Nikolai
S. Zefirovb
“A.N. Nesmeyanov Institute of Elementoorganic Compounds, Russian Academy of Science, Vavilov str. 28, Moscow 117813. Russia bN.D. Zelinsky Institute of OrganicChemistry, Russian Academy of Science, Leninsky Pr. 47, Moscow 117913, Russia
Received 6 January 1995;accepted 4 February 1995
Abstract The topological indices (TIs)-invariants of weighted molecular graphs representing the chemical structures-are widely used as molecular descriptors in quantitative structure-property relationships (QSPR). The essential disadvantage of the TIs-method in QSPR is the absence of any physical-chemical interpretation of the majority of TIs used in QSPR. This leads to problems in the choice of TIs and justification of the obtained correlations, the “TIs-property”. In the present paper a quantum chemical interpretation of connectivity TIs widely used in QSPR for some classes of conjugated hydrocarbons is suggested. The results obtained justify using these TIs in QSPR for such compounds. Keywords:
Hydrocarbon; Molecular connectivity indices; QSPR
1. Introduction Investigations connected with the construction and application of topological indices (TIs) are becoming increasingly popular in mathematical chemistry [l-3]. The TIs are defined as numerical invariants of weighted molecular graphs, representing the chemical structural formulae. The TIs may be used for the codification, ordering and searching for chemical structures in databases, and for solving the graph isomorphism problem. However the main field of their application is in QSPR (quantitative structure-property relationships) where TIs are used as the descriptors of molecular structures. The most popular of all TIs used in QSPR is the * Corresponding author.
Randic connectivity index x initially for ordering acyclic structures [l-4]:
introduced
(1) edges
(iA
where Vi and Uj are the degrees of the ith and jth graph vertices. There are different generalizations of index x for heteroatomic systems (valenceweighted connectivity indices) and for cases when the sum in Eq. (1) runs over more complicated subgraphs of the molecular graph (connectivity indices of higher orders). The literature contains many correlations of connectivity indices with various properties for different classes of chemical compounds [l-4]. It is natural to define the following generalization of index x, index Xd, and to call it “the
0166-1280/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0166-1280(95)04187-7
174
I. V. Stankevich et al./.lournal of Molecular Structure (Theochem)
generalized Randic index of the first order”: Xd
=
c (XiXjjdbij edges
(4
(iJ)
where xi and xj are some weights of the ith and jth graph vertices, bij is a weight of the edge (i, j) and d is a constant. Obviously, Xd = x, if d = -l/2, xi = vi, xj = v_/, bij = 1. Particular cases of index Xd with definite parameters Xi, xj, bij and d have also been considered in QSPR [l-3]. The TIs-approach in QSPR has a number of advantages and disadvantages. Its main advantage is the possibility of calculation of TIs in many different ways directly from structural formulae, using simple calculations. The possibility of including in TIs some geometrical and electronic characteristics of molecules should also be noted. The essential disadvantage of the TIs-method, mentioned already in a review [l], is the absence of any physical-chemical interpretation of the majority of TIs constructed using refined mathematical operations with graphs. This formalism in TIs-methods leads to problems in the choice of TIs and justification of the equations obtained in QSPR. Without such justification the procedure of constructing the correlations, “TIs-property”, turns into a “game with numbers”, and it is very possible to get random correlations. However, a number of TIs can be interpreted in a framework of some physical or chemical theory. For example, such TIs as the numbers of simple structural fragments of a molecule or TIs of spectral type, which appear in quantum chemical methods, have a physical-chemical meaning [l]. The existence of the relationships between these TIs and other, formally constructed TIs allows us to interpret the latter as well. So, establishing such relationships is an important problem in QSPR studies. In an earlier work [5] we have constructed the problem of quantum chemical interpretation of indices x and Xd for two classes of conjugated hydrocarbons-cata-condensed benzenoid hydrocarbons and polyphenylenes. It was shown that the generalized Randic index Xd with some weights xi, b, coincides with some energy functional (Hamiltonian function) describing quantum
342 (1995)
173-l 79
chemical properties of such molecules. The minimum of this functional is equal to the total 7r-electron energy of the system and it is defined by an expression depending on the index x and the number of vertices in the molecular graph. In the present paper the results of Ref. [5] are extended to other classes of conjugated hydrocarbons.
2. Main results Let us consider the following classes K,--K, of conjugated hydrocarbons. K,, KZ, K3: Cata-condensed benzenoid hydrocarbons consisting of 6-membered, 5-membered, or 7-membered rings, accordingly (Figs. l(a)-l(c)). K4: Polycyclic hydrocarbons consisting of 6-membered rings connected one with another by edges (polyphenylenes). Every ring can be connected with 1, 2 or 3 rings only. Every two vertices of a ring, to which other rings are attached, are non-adjoint vertices. A graph has the structure of a tree, that is it does not contain other rings, besides those mentioned above (Fig. l(d)). K5,c: Peri-condensed benzenoid hydrocarbons consisting of 6-membered rings. We denote by xi the numbers of rings in such graphs with i adjoint rings, i = 1, . . ,, 6. Suppose that for each ring those edges belonging only to it form a connected edge chain. Denote by K5,c such graphs for which xl = 0 and 2x2 + x3 = c, where c is a fixed constant (Fig. l(e), c = 6). Kg: Molecular graphs of chemical compounds of this class consist of two simple rings connected by a chain of arbitrary length k,, > 1. The numbers of edges k, and k2 in the rings can be equal to 5, 6, or 7 (Fig. l(f)). K7: Molecular graphs of compounds of this class are trees with vertex degrees 1, 2 or 3 and the additional restrictions e22 = 0, e33 = 0, where eii (i = 2, 3) is equal to the number of edges connecting two vertices of the ith degree (Fig. l(g)). In this paper we shall demonstrate the following points. (a) For molecular graphs of classes K,-K, the generalized Randic index Xd with d = l/2 and some weights xi, bij coincides with some energy
I. V. Srankevich et al./Journal of Molecular Structure (Theochem)
KI
342 (1995)
173- -179
175
K2
(b)
e@ (y&J K4
Q
(cd
r=3
r=4
(dl
r=5 (e)
Fig. 1. Examples of molecular graphs for classes K,-K,.
functional (Hamilton function) describing the ?r-electron properties of the corresponding conjugated molecules. The minimum of this functional is equal to the total r-electron energy of the system and is defined by an expression depending on two TIs-the Randic index x and the number of vertices in the molecular graph. (b) For graphs of classes KI-K7 there are
relationships between TIs such as: (1) Xd - x(d= -l/2, Xi = 0i, bii = 1); (2) Xd (d = 1, Xi = ‘pi, bi, = 1); (3) E, (total r-electron energy calculated by the Htickel approach: (4) A,,, (maximal eigenvalue of the graph); (5) cy= 1uf (vi is the degree of the ith graph vertex);
I. V.
176
Stankevich
et al.lJournal
of Molecular
(6) n (number of vertices in the graph); namely, the following approximate formulas are true:
=
nk
aln + 4x +
a3
( a@ + a5x f 06 )
(3)
where Ui (i= 1, . . .. 6) and k are some constants dependent on the class of compounds; k is an empirical parameter defined by statistical methods; Qi (i= 1, ...) 6) are analytically calculated values. Let us turn to the proof of the statements given above. It is well known that for investigation of electronic structures of complicated systems different mathematical models defined by the energy functional H(B) (where 0 is a many-electron wave function) and the set of basic one-electron atomic orbitals 4,). . . ,&,, are used [6,7]. In the LCAO MO approximation for systems with a closed electronic shell the energy functional H(0) turn into the many variable function h(. . , cii, . .), depending on the coefficients cij in expansions of MO 0i on basic orbitals; 8i = Cy= 1~jcij, (j = 1, . . . , m; i= l,..., n’; 2n’ is equal to the number of electrons). The numbers Cij are defined from the energy minimum under conditions of orthonormalization of MO. The expression for h(. . . , tij,. . .) can be also written as a function of the density matrix, P = IIPijll, i, j = 1, . ., m. The matrix P is square symmetric and its elements are calculated in terms of cij. If the number of electrons is even, then P2 = 2P, Ci Pii = 2n’. Hence, the matrix elements Pi, are dependent [7]. From the results of paper [S], which established a discrete analog of the Hoenberg-Kohn theorem for systems with even numbers of electrons [9], is follows that it is possible to write the energy function h(. . . , Cij, . . .) as a function depending on only the diagonal elements of matrix P. Such an energy form for r-electron systems of conjugated hydrocarbons may be described by the following function [IO]: h(q,,...,qn)
=P
C
k;i(qiqj)1’2 +on
edges(i,j)
where n is the number of atoms forming the conjugated part of the molecule, o is the Coulombic
Structure
(Theorhem)
342 (1995)
!73-
179
integral, p is the resonance integral; CY,0, kij are empirical parameters depending on the class of chemical compounds and bond types, qi; = Pi, is the 7r-electron density on the ith atom of the conjugated system, C qi = n. The minimum of the the conditions function h(q,, . . . ,q,l) (under C qi = n, qj > 0) is equal t0 E = min h(qi, . . , qn) = na f n/3X&, Ys
where XL,, is the maximal eigenvalue of the edgeweighted molecular graph describing the conjugated part of the molecule; the weight of edge (i, j) is equal to ki,j. In some approximations one can suppose that parameters kij are the same for all indices i and j, that is k = kij. It was established for conjugated hydrocarbons of a sufficiently broad class, that if k;j = k = 0.56, then E x E,, where E, is the total r-electron energy calculated using the Htickel method [ll]. Let us take (Y= 0, p = 1. Then E in such a normalized system of units can be written as E z E, = nkX,,,, where X,,, is the maximal eigenvalue of a non-weighted molecular graph corresponding to the conjugated part of the molecule. Obviously, the function h(q,, . . . , qn) (in normalized form) coincides with the generalized Randic index Xd, where d = l/2, Xi = qi, bij = ,Bkij = k,,. Let us now deduce the approximate relationships between E, x1 (at xi = v;, bij = I), A,,, and Cr=, II’. For this purpose it is necessary to estimate X,,,. From the variation principle for eigenvalues, using as the trial vector c = (ZQ, . . . , Us), we find that (see also Ref. [l 11):
= 2x1 (cl,:)-’ Further we will consider the classes K,-K7 separately. Denote by eij the number of edges connecting vertices of degree i and j (i
I. V. Stankevich et al./Journal of Molecular Structure ( Theochem) 342 (1995)
Let us consider molecular is easy to see that e33 =ei3+r--
1,
graphs
of class
K1.It
173-I 79
Let us consider molecular is easy to see that
graphs
e23 = -2ei3
e22 = e& + r + 6,
e23 = 4r - 4 - 2ei3,
177
of class K3_ It - 4,
$4r
e33 = e& + r - 1 e22
=
eIi3
+
di
6,
=
f
(e23
+
2eii), i = 2,3; IlilJj = e& + 37r - 9,
X1 = C
x71:
= 30r - 2,
x = ies3 +Je22 Cl./)
Xl = C
n=5r+2
‘;)
=ei3(5f__‘2)+i-($)+(8Ti ViUj = e& + 33r - 9,
edges (f.il
c
uf
26r - 2,
=
n=4r+2 Hence,
Hence,
xmax = z
x( 120 + 48&)
- n(24&
eg3 =r-
e22 = 4,
1,
d3 = 2e’j3 = 2r - 2,
graphs
d2=r+4
+
XI
c
=
ViVj
=
4e22+ 9e33+ 6e23 = 30r - 11
(i.i) II?
=
=
=
3v%
x(33&
- 180) - 9Ov% + 158 - 66) - 33&
)
9e33+ 6e13+ 4e22 = 41r
x max =
Hence.
x max
e2? =2r+4
l),
34r - 10,
-
17,
n = 6r
Hence,
4d2 + 9d, = 22r - 2
x(90&
of class Kq. It
&,A-12
c vf=
edges
c
ez3 =4(r-
graphs
e23 = 2r
( XI
- 133
15n - 35
e33 = r - 1,
of class K2. It
ei3 = r - 2,
- ~(36 + 30&j
Let us consider molecular is easy to see, that
+ 25) - (114 + 40&)
13n - 30
Let us consider molecular is easy to see that
x(150 + 60&)
e23
=
x(51&-
- 834)
102) - (153&-
354)
Let us consider molecular graphs of class KS,c. In this case the following equations are true:
+ 64
e22 = 2x2 + x3 = c,
x( 123v’6 - 246) - (369&
x2 + x3 +
2x2 + 2x3 + 2x4,
x4
+
x5
+
x6
=
r,
e33
e22 + e23 + e& + 2e’js = 6r,
ef(3 = i (6x6 + 5x, + 4x4 + 3x3 + 2x2)
=
ei3
+
ei3 = x5
eli3
I. V. Stankevich et al.l.Iournal of Molecular Structure (Theochem) 342 (1995) 173-l 79
178
It follows from these equations that:
Let us consider molecular graphs of class K7. It is easy to prove that
x=
di =ei2+ei3,
;e22+fe,,+A,,=x, &
2d2 = ei2 + e23,
3d3 = ei3 + e23,
dl = d3 + 2
Besides, e23 + e22 = 3x2 + 2X3 + x4
d2 = 2
d3=~+je,3=~X2+tX3+2Xq+fXg+2Xg
c
n = dz + d3
= 3 (6x6 + 7xs + 9x4 + 11x3 + 13x2) f (6r + x5 + 3x4 + 5x3 + 7x2) Vivj =
G
29x2 +
29.5~3 + 30x4 + 31.5~5 + 27x6
edges (U 1
n = d, + d2 + d3 c
vf=d,+4d2+9d3=2d3+4n-6
Hence, x max
?I; =
c
~iwj= 2e12 + 3e13 + 6ez3 = 5n - 11,
edges (iJ 1
24x2
+ 23x3
+
22x4 + 21x5 + 18~~
x2 +x3 +x4 = (6& + 15)n - (12& + 30)x
5n- 11 M d3 + 2n - 3 n(20JZ-l5J5--5)-44fi+33&+11
So,
=x&+n(9&-7fi--3)-15&+10&+6
x max 54r + (4x2 -I 5x3 + 6x4 + 9x5)
So, the formulas obtained.
(3) for classes Ki-K7
are
= 18r + (6x2 + 5x3 + 4x4 + 3x5) x(630 + 252&) =
- n(288 + 126&) - 19~
x( 150 - 60&) - n(66 - 30&) - 5c
Let us consider molecular graphs of class Kg. It is easy to see that e33 = 0,
e23 = 6,
e22 = kl + k2 + ko - 6
x=;(k,+k2+ko)-3+&, n=k,+k2+ko-1 C
tiiuj = 4(kl + k2 + k,) + 12,
edges (hi)
Cv:=6+4(kl+kz+ko)
Hence, x max
8% + 24 - 8x&
= 4x+9-4&
3. Conclusions (1) The generalized Randic index x1i2 for conjugated hydrocarbons of classes Ki -K7 can be interpreted as the energy functional h(ql, . . . , q,) depending on the rr-electron density on the atoms of the corresponding molecules. (2) The minimum E of this functional is a function of the Randic index x and the number of graph vertices n: E M bin + hx + b3 b4n + bSx + b6
where bi are some constants depending on the class of compounds. This fact allows us to interpret the index x as the fundamental energy characteristic of the conjugated system and to justify the use
I. V. Stankevich et aLlJournal of Molecular Structure (Theochem)
of x in QSPR-studies for conjugated hydrocarbons. Hence, the Randic index defined formally for ordering acyclic structures [4], appears in a natural way in the quantum chemistry theory of conjugated molecules. (3) It is known that the index x for acyclic graphs can be considered as the degree of branching of the corresponding structure. The reason is that the minimal and maximal values of x (between all acyclic graphs with a fixed number of vertices) correspond to chain-graph and star-graph, which are the most and the least highly branched structures, from an intuitive point of view. However, as we have proved above, for graphs from classes Kz, K4 and Kg, the index x depends on the number of vertices n only. Hence, for isomeric structures of such classes x takes one and the same value and does not reflect the degree of branching. (4) Since relationships between the topological indices E,, E, x-1/2, ~1 (xi = Vi, bij = I), A,,, and n exist, the parameters xl and A,,, may also be interpreted as useful energy characteristics of 7r-electron systems.
4. Remarks (1) Formulas connecting E, and x, similar to (3) may also be obtained in the same way for some other classes of conjugated hydrocarbons. For example, one can take any two graphs from K, -K7 connect one with another by an edge chain, and in this way produce a new class of molecular graphs. (2) Results similar to those discussed in the present paper for index x, may be obtained for connectivity indices of higher orders. For this purpose it is necessary to take some other functional h(q,, . . , qn) in the first step of the investigations. (3) In Ref. [12] a regression equation of the kind E,=ax+b (a, b are some constants)
(4)
was obtained for a set consisting of 60 conjugated hydrocarbons. The differences between our results and those
342 (1995) 173-l 79
179
mentioned above are the following. The dependence E, = E,(x) in Ref. [12] is suggested a priori to be linear; (4) contains two empirical parameters, a and b; these parameters are calculated using regression analysis for a definite set of compounds. In our approach the non-linear dependence E, = E,(x, n) is deduced analytically, for large classes of compounds described in terms of structural features; (3) contains only one empirical parameter, k. It follows from the statistical analysis that for classes K1-K7 0.52 6 k ~0.6.
Acknowledgments
We thank Igor Baskin and Olga Slovokhotova for their help in creating a computer chemical database and data processing, necessary for statistical calculation of the constant k.
References [l] M.I. Stankevich, I.V. Stankevich and N.S. Zefirov, Usp. Khim., 57 (1988) 337. [2] D. Rouvray, in R.B. King (Ed.), Chemical Applications of Topology and Graph Theory, translated into Russian, Izd. Mir, Moscow, 1987, p. 183. [3] L.B. Kier and L.H. Hall, Molecular Connectivity in Structure-Activity Analysis, Research Studies Press, Letchworth, U.K., 1986. [4] M. Randic, J. Am. Chem. Sot., 97 (1975) 6609. [S] I.V. Stankevich, MI. Skvortsova and N.S. Zefirov. Dokl. Akad. Nauk, 324 (1992) 133. [6] K. Higasi, H. Baba and A. Rembaum, Quantum Organic Chemistry, translated into Russian, Izd. Mir, Moscow, 1967. [7] M.M. Mestetchkin, Method of Density Matrix in Theory of Molecules, Izd. Naukova Dumka, Kiev, 1977 (in Russian). [8] I.V. Stankevich and R.Sh. Baquradze, Zh. Fiz. Khim., 9 (1986) 1924. [9] R. Hohenberg and W. Kohn, Phys. Rev. B, 136 (1964) 864. [IO] M.V. Nikerov and I.V. Stankevich, Zh. Strukt. Khim., 22 (1981) 3. [ll] I.V. Stankevich and R.Sh. Baquradze, Izv. Akad. Nauk Gruzinskoi SSR, Ser. Khim., 14 (1988) 217. [12] M. Randic, Z. Jericevic, A. Sablic and N. Trinajstic, Acta Phys. Pol. Ser. A, 74 (1988) 317.
On a quantum chemical interpretation of molecular connectivity indices for conjugated hydrocarbons Ivan V. Stankevich”>*,
Mariya
I. Skvortsovab,
Nikolai
S. Zefirovb
“A.N. Nesmeyanov Institute of Elementoorganic Compounds, Russian Academy of Science, Vavilov str. 28, Moscow 117813. Russia bN.D. Zelinsky Institute of OrganicChemistry, Russian Academy of Science, Leninsky Pr. 47, Moscow 117913, Russia
Received 6 January 1995;accepted 4 February 1995
Abstract The topological indices (TIs)-invariants of weighted molecular graphs representing the chemical structures-are widely used as molecular descriptors in quantitative structure-property relationships (QSPR). The essential disadvantage of the TIs-method in QSPR is the absence of any physical-chemical interpretation of the majority of TIs used in QSPR. This leads to problems in the choice of TIs and justification of the obtained correlations, the “TIs-property”. In the present paper a quantum chemical interpretation of connectivity TIs widely used in QSPR for some classes of conjugated hydrocarbons is suggested. The results obtained justify using these TIs in QSPR for such compounds. Keywords:
Hydrocarbon; Molecular connectivity indices; QSPR
1. Introduction Investigations connected with the construction and application of topological indices (TIs) are becoming increasingly popular in mathematical chemistry [l-3]. The TIs are defined as numerical invariants of weighted molecular graphs, representing the chemical structural formulae. The TIs may be used for the codification, ordering and searching for chemical structures in databases, and for solving the graph isomorphism problem. However the main field of their application is in QSPR (quantitative structure-property relationships) where TIs are used as the descriptors of molecular structures. The most popular of all TIs used in QSPR is the * Corresponding author.
Randic connectivity index x initially for ordering acyclic structures [l-4]:
introduced
(1) edges
(iA
where Vi and Uj are the degrees of the ith and jth graph vertices. There are different generalizations of index x for heteroatomic systems (valenceweighted connectivity indices) and for cases when the sum in Eq. (1) runs over more complicated subgraphs of the molecular graph (connectivity indices of higher orders). The literature contains many correlations of connectivity indices with various properties for different classes of chemical compounds [l-4]. It is natural to define the following generalization of index x, index Xd, and to call it “the
0166-1280/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0166-1280(95)04187-7
174
I. V. Stankevich et al./.lournal of Molecular Structure (Theochem)
generalized Randic index of the first order”: Xd
=
c (XiXjjdbij edges
(4
(iJ)
where xi and xj are some weights of the ith and jth graph vertices, bij is a weight of the edge (i, j) and d is a constant. Obviously, Xd = x, if d = -l/2, xi = vi, xj = v_/, bij = 1. Particular cases of index Xd with definite parameters Xi, xj, bij and d have also been considered in QSPR [l-3]. The TIs-approach in QSPR has a number of advantages and disadvantages. Its main advantage is the possibility of calculation of TIs in many different ways directly from structural formulae, using simple calculations. The possibility of including in TIs some geometrical and electronic characteristics of molecules should also be noted. The essential disadvantage of the TIs-method, mentioned already in a review [l], is the absence of any physical-chemical interpretation of the majority of TIs constructed using refined mathematical operations with graphs. This formalism in TIs-methods leads to problems in the choice of TIs and justification of the equations obtained in QSPR. Without such justification the procedure of constructing the correlations, “TIs-property”, turns into a “game with numbers”, and it is very possible to get random correlations. However, a number of TIs can be interpreted in a framework of some physical or chemical theory. For example, such TIs as the numbers of simple structural fragments of a molecule or TIs of spectral type, which appear in quantum chemical methods, have a physical-chemical meaning [l]. The existence of the relationships between these TIs and other, formally constructed TIs allows us to interpret the latter as well. So, establishing such relationships is an important problem in QSPR studies. In an earlier work [5] we have constructed the problem of quantum chemical interpretation of indices x and Xd for two classes of conjugated hydrocarbons-cata-condensed benzenoid hydrocarbons and polyphenylenes. It was shown that the generalized Randic index Xd with some weights xi, b, coincides with some energy functional (Hamiltonian function) describing quantum
342 (1995)
173-l 79
chemical properties of such molecules. The minimum of this functional is equal to the total 7r-electron energy of the system and it is defined by an expression depending on the index x and the number of vertices in the molecular graph. In the present paper the results of Ref. [5] are extended to other classes of conjugated hydrocarbons.
2. Main results Let us consider the following classes K,--K, of conjugated hydrocarbons. K,, KZ, K3: Cata-condensed benzenoid hydrocarbons consisting of 6-membered, 5-membered, or 7-membered rings, accordingly (Figs. l(a)-l(c)). K4: Polycyclic hydrocarbons consisting of 6-membered rings connected one with another by edges (polyphenylenes). Every ring can be connected with 1, 2 or 3 rings only. Every two vertices of a ring, to which other rings are attached, are non-adjoint vertices. A graph has the structure of a tree, that is it does not contain other rings, besides those mentioned above (Fig. l(d)). K5,c: Peri-condensed benzenoid hydrocarbons consisting of 6-membered rings. We denote by xi the numbers of rings in such graphs with i adjoint rings, i = 1, . . ,, 6. Suppose that for each ring those edges belonging only to it form a connected edge chain. Denote by K5,c such graphs for which xl = 0 and 2x2 + x3 = c, where c is a fixed constant (Fig. l(e), c = 6). Kg: Molecular graphs of chemical compounds of this class consist of two simple rings connected by a chain of arbitrary length k,, > 1. The numbers of edges k, and k2 in the rings can be equal to 5, 6, or 7 (Fig. l(f)). K7: Molecular graphs of compounds of this class are trees with vertex degrees 1, 2 or 3 and the additional restrictions e22 = 0, e33 = 0, where eii (i = 2, 3) is equal to the number of edges connecting two vertices of the ith degree (Fig. l(g)). In this paper we shall demonstrate the following points. (a) For molecular graphs of classes K,-K, the generalized Randic index Xd with d = l/2 and some weights xi, bij coincides with some energy
I. V. Srankevich et al./Journal of Molecular Structure (Theochem)
KI
342 (1995)
173- -179
175
K2
(b)
e@ (y&J K4
Q
(cd
r=3
r=4
(dl
r=5 (e)
Fig. 1. Examples of molecular graphs for classes K,-K,.
functional (Hamilton function) describing the ?r-electron properties of the corresponding conjugated molecules. The minimum of this functional is equal to the total r-electron energy of the system and is defined by an expression depending on two TIs-the Randic index x and the number of vertices in the molecular graph. (b) For graphs of classes KI-K7 there are
relationships between TIs such as: (1) Xd - x(d= -l/2, Xi = 0i, bii = 1); (2) Xd (d = 1, Xi = ‘pi, bi, = 1); (3) E, (total r-electron energy calculated by the Htickel approach: (4) A,,, (maximal eigenvalue of the graph); (5) cy= 1uf (vi is the degree of the ith graph vertex);
I. V.
176
Stankevich
et al.lJournal
of Molecular
(6) n (number of vertices in the graph); namely, the following approximate formulas are true:
=
nk
aln + 4x +
a3
( a@ + a5x f 06 )
(3)
where Ui (i= 1, . . .. 6) and k are some constants dependent on the class of compounds; k is an empirical parameter defined by statistical methods; Qi (i= 1, ...) 6) are analytically calculated values. Let us turn to the proof of the statements given above. It is well known that for investigation of electronic structures of complicated systems different mathematical models defined by the energy functional H(B) (where 0 is a many-electron wave function) and the set of basic one-electron atomic orbitals 4,). . . ,&,, are used [6,7]. In the LCAO MO approximation for systems with a closed electronic shell the energy functional H(0) turn into the many variable function h(. . , cii, . .), depending on the coefficients cij in expansions of MO 0i on basic orbitals; 8i = Cy= 1~jcij, (j = 1, . . . , m; i= l,..., n’; 2n’ is equal to the number of electrons). The numbers Cij are defined from the energy minimum under conditions of orthonormalization of MO. The expression for h(. . . , tij,. . .) can be also written as a function of the density matrix, P = IIPijll, i, j = 1, . ., m. The matrix P is square symmetric and its elements are calculated in terms of cij. If the number of electrons is even, then P2 = 2P, Ci Pii = 2n’. Hence, the matrix elements Pi, are dependent [7]. From the results of paper [S], which established a discrete analog of the Hoenberg-Kohn theorem for systems with even numbers of electrons [9], is follows that it is possible to write the energy function h(. . . , Cij, . . .) as a function depending on only the diagonal elements of matrix P. Such an energy form for r-electron systems of conjugated hydrocarbons may be described by the following function [IO]: h(q,,...,qn)
=P
C
k;i(qiqj)1’2 +on
edges(i,j)
where n is the number of atoms forming the conjugated part of the molecule, o is the Coulombic
Structure
(Theorhem)
342 (1995)
!73-
179
integral, p is the resonance integral; CY,0, kij are empirical parameters depending on the class of chemical compounds and bond types, qi; = Pi, is the 7r-electron density on the ith atom of the conjugated system, C qi = n. The minimum of the the conditions function h(q,, . . . ,q,l) (under C qi = n, qj > 0) is equal t0 E = min h(qi, . . , qn) = na f n/3X&, Ys
where XL,, is the maximal eigenvalue of the edgeweighted molecular graph describing the conjugated part of the molecule; the weight of edge (i, j) is equal to ki,j. In some approximations one can suppose that parameters kij are the same for all indices i and j, that is k = kij. It was established for conjugated hydrocarbons of a sufficiently broad class, that if k;j = k = 0.56, then E x E,, where E, is the total r-electron energy calculated using the Htickel method [ll]. Let us take (Y= 0, p = 1. Then E in such a normalized system of units can be written as E z E, = nkX,,,, where X,,, is the maximal eigenvalue of a non-weighted molecular graph corresponding to the conjugated part of the molecule. Obviously, the function h(q,, . . . , qn) (in normalized form) coincides with the generalized Randic index Xd, where d = l/2, Xi = qi, bij = ,Bkij = k,,. Let us now deduce the approximate relationships between E, x1 (at xi = v;, bij = I), A,,, and Cr=, II’. For this purpose it is necessary to estimate X,,,. From the variation principle for eigenvalues, using as the trial vector c = (ZQ, . . . , Us), we find that (see also Ref. [l 11):
= 2x1 (cl,:)-’ Further we will consider the classes K,-K7 separately. Denote by eij the number of edges connecting vertices of degree i and j (i
I. V. Stankevich et al./Journal of Molecular Structure ( Theochem) 342 (1995)
Let us consider molecular is easy to see that e33 =ei3+r--
1,
graphs
of class
K1.It
173-I 79
Let us consider molecular is easy to see that
graphs
e23 = -2ei3
e22 = e& + r + 6,
e23 = 4r - 4 - 2ei3,
177
of class K3_ It - 4,
$4r
e33 = e& + r - 1 e22
=
eIi3
+
di
6,
=
f
(e23
+
2eii), i = 2,3; IlilJj = e& + 37r - 9,
X1 = C
x71:
= 30r - 2,
x = ies3 +Je22 Cl./)
Xl = C
n=5r+2
‘;)
=ei3(5f__‘2)+i-($)+(8Ti ViUj = e& + 33r - 9,
edges (f.il
c
uf
26r - 2,
=
n=4r+2 Hence,
Hence,
xmax = z
x( 120 + 48&)
- n(24&
eg3 =r-
e22 = 4,
1,
d3 = 2e’j3 = 2r - 2,
graphs
d2=r+4
+
XI
c
=
ViVj
=
4e22+ 9e33+ 6e23 = 30r - 11
(i.i) II?
=
=
=
3v%
x(33&
- 180) - 9Ov% + 158 - 66) - 33&
)
9e33+ 6e13+ 4e22 = 41r
x max =
Hence.
x max
e2? =2r+4
l),
34r - 10,
-
17,
n = 6r
Hence,
4d2 + 9d, = 22r - 2
x(90&
of class Kq. It
&,A-12
c vf=
edges
c
ez3 =4(r-
graphs
e23 = 2r
( XI
- 133
15n - 35
e33 = r - 1,
of class K2. It
ei3 = r - 2,
- ~(36 + 30&j
Let us consider molecular is easy to see, that
+ 25) - (114 + 40&)
13n - 30
Let us consider molecular is easy to see that
x(150 + 60&)
e23
=
x(51&-
- 834)
102) - (153&-
354)
Let us consider molecular graphs of class KS,c. In this case the following equations are true:
+ 64
e22 = 2x2 + x3 = c,
x( 123v’6 - 246) - (369&
x2 + x3 +
2x2 + 2x3 + 2x4,
x4
+
x5
+
x6
=
r,
e33
e22 + e23 + e& + 2e’js = 6r,
ef(3 = i (6x6 + 5x, + 4x4 + 3x3 + 2x2)
=
ei3
+
ei3 = x5
eli3
I. V. Stankevich et al.l.Iournal of Molecular Structure (Theochem) 342 (1995) 173-l 79
178
It follows from these equations that:
Let us consider molecular graphs of class K7. It is easy to prove that
x=
di =ei2+ei3,
;e22+fe,,+A,,=x, &
2d2 = ei2 + e23,
3d3 = ei3 + e23,
dl = d3 + 2
Besides, e23 + e22 = 3x2 + 2X3 + x4
d2 = 2
d3=~+je,3=~X2+tX3+2Xq+fXg+2Xg
c
n = dz + d3
= 3 (6x6 + 7xs + 9x4 + 11x3 + 13x2) f (6r + x5 + 3x4 + 5x3 + 7x2) Vivj =
G
29x2 +
29.5~3 + 30x4 + 31.5~5 + 27x6
edges (U 1
n = d, + d2 + d3 c
vf=d,+4d2+9d3=2d3+4n-6
Hence, x max
?I; =
c
~iwj= 2e12 + 3e13 + 6ez3 = 5n - 11,
edges (iJ 1
24x2
+ 23x3
+
22x4 + 21x5 + 18~~
x2 +x3 +x4 = (6& + 15)n - (12& + 30)x
5n- 11 M d3 + 2n - 3 n(20JZ-l5J5--5)-44fi+33&+11
So,
=x&+n(9&-7fi--3)-15&+10&+6
x max 54r + (4x2 -I 5x3 + 6x4 + 9x5)
So, the formulas obtained.
(3) for classes Ki-K7
are
= 18r + (6x2 + 5x3 + 4x4 + 3x5) x(630 + 252&) =
- n(288 + 126&) - 19~
x( 150 - 60&) - n(66 - 30&) - 5c
Let us consider molecular graphs of class Kg. It is easy to see that e33 = 0,
e23 = 6,
e22 = kl + k2 + ko - 6
x=;(k,+k2+ko)-3+&, n=k,+k2+ko-1 C
tiiuj = 4(kl + k2 + k,) + 12,
edges (hi)
Cv:=6+4(kl+kz+ko)
Hence, x max
8% + 24 - 8x&
= 4x+9-4&
3. Conclusions (1) The generalized Randic index x1i2 for conjugated hydrocarbons of classes Ki -K7 can be interpreted as the energy functional h(ql, . . . , q,) depending on the rr-electron density on the atoms of the corresponding molecules. (2) The minimum E of this functional is a function of the Randic index x and the number of graph vertices n: E M bin + hx + b3 b4n + bSx + b6
where bi are some constants depending on the class of compounds. This fact allows us to interpret the index x as the fundamental energy characteristic of the conjugated system and to justify the use
I. V. Stankevich et aLlJournal of Molecular Structure (Theochem)
of x in QSPR-studies for conjugated hydrocarbons. Hence, the Randic index defined formally for ordering acyclic structures [4], appears in a natural way in the quantum chemistry theory of conjugated molecules. (3) It is known that the index x for acyclic graphs can be considered as the degree of branching of the corresponding structure. The reason is that the minimal and maximal values of x (between all acyclic graphs with a fixed number of vertices) correspond to chain-graph and star-graph, which are the most and the least highly branched structures, from an intuitive point of view. However, as we have proved above, for graphs from classes Kz, K4 and Kg, the index x depends on the number of vertices n only. Hence, for isomeric structures of such classes x takes one and the same value and does not reflect the degree of branching. (4) Since relationships between the topological indices E,, E, x-1/2, ~1 (xi = Vi, bij = I), A,,, and n exist, the parameters xl and A,,, may also be interpreted as useful energy characteristics of 7r-electron systems.
4. Remarks (1) Formulas connecting E, and x, similar to (3) may also be obtained in the same way for some other classes of conjugated hydrocarbons. For example, one can take any two graphs from K, -K7 connect one with another by an edge chain, and in this way produce a new class of molecular graphs. (2) Results similar to those discussed in the present paper for index x, may be obtained for connectivity indices of higher orders. For this purpose it is necessary to take some other functional h(q,, . . , qn) in the first step of the investigations. (3) In Ref. [12] a regression equation of the kind E,=ax+b (a, b are some constants)
(4)
was obtained for a set consisting of 60 conjugated hydrocarbons. The differences between our results and those
342 (1995) 173-l 79
179
mentioned above are the following. The dependence E, = E,(x) in Ref. [12] is suggested a priori to be linear; (4) contains two empirical parameters, a and b; these parameters are calculated using regression analysis for a definite set of compounds. In our approach the non-linear dependence E, = E,(x, n) is deduced analytically, for large classes of compounds described in terms of structural features; (3) contains only one empirical parameter, k. It follows from the statistical analysis that for classes K1-K7 0.52 6 k ~0.6.
Acknowledgments
We thank Igor Baskin and Olga Slovokhotova for their help in creating a computer chemical database and data processing, necessary for statistical calculation of the constant k.
References [l] M.I. Stankevich, I.V. Stankevich and N.S. Zefirov, Usp. Khim., 57 (1988) 337. [2] D. Rouvray, in R.B. King (Ed.), Chemical Applications of Topology and Graph Theory, translated into Russian, Izd. Mir, Moscow, 1987, p. 183. [3] L.B. Kier and L.H. Hall, Molecular Connectivity in Structure-Activity Analysis, Research Studies Press, Letchworth, U.K., 1986. [4] M. Randic, J. Am. Chem. Sot., 97 (1975) 6609. [S] I.V. Stankevich, MI. Skvortsova and N.S. Zefirov. Dokl. Akad. Nauk, 324 (1992) 133. [6] K. Higasi, H. Baba and A. Rembaum, Quantum Organic Chemistry, translated into Russian, Izd. Mir, Moscow, 1967. [7] M.M. Mestetchkin, Method of Density Matrix in Theory of Molecules, Izd. Naukova Dumka, Kiev, 1977 (in Russian). [8] I.V. Stankevich and R.Sh. Baquradze, Zh. Fiz. Khim., 9 (1986) 1924. [9] R. Hohenberg and W. Kohn, Phys. Rev. B, 136 (1964) 864. [IO] M.V. Nikerov and I.V. Stankevich, Zh. Strukt. Khim., 22 (1981) 3. [ll] I.V. Stankevich and R.Sh. Baquradze, Izv. Akad. Nauk Gruzinskoi SSR, Ser. Khim., 14 (1988) 217. [12] M. Randic, Z. Jericevic, A. Sablic and N. Trinajstic, Acta Phys. Pol. Ser. A, 74 (1988) 317.