On π-electron configuration of cyclopenta-derivatives of benzenoid hydrocarbons
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1 Indian Journal of Chemistry Vol. 49, July 010, pp On π-electron configuration of cyclopenta-derivatives of benzenoid hydrocarbons Ivan Gutman*, Jelena Đurđević, Dušan Bašić & Dragana Rašović Faculty of Science, P.O. Box 60, Kragujevac, Serbia Received 19 May 010; accepted 15 June 010 Within the Hückel molecular orbital model, Kekulèan benzenoid hydrocarbons have equal number of bonding and antibonding molecular orbitals, and no non-bonding molecular orbitals. In cyclopenta-derivatives of benzenoid hydrocarbons this regularity may be either preserved or violated, depending on the number and position of the fivemembered rings. Some general regularities along these lines are established. In particular, the problem is completely solved in the case of prolate-rectangle-shaped benzenoids; the number of bonding molecular orbitals always exceeds the number of antibonding molecular orbitals, except if all five-membered rings lie on the same side of the benzenoid system. Keywords: Theoretical chemistry, Graph theory, Electron configuration, Molecular orbitals, Hydrocarbons, Benzenoids In an attempt to extend the earlier well developed theory of benzenoid hydrocarbons 1-5 to their derivatives containing five-membered rings, we have recently initiated a systematic study of the π-electron properties of such species containing one 6-14 or two cyclopentadiene fragments. lready in the case of dicyclopenta-derivatives, we encountered pronounced irregularities 15,16 caused by the fact that these have more bonding than antibonding molecular orbitals (MOs), and therefore some of their bonding MOs are unoccupied. The natural question that arises at this point is what happens in the general case when the benzenoid derivatives possess more than one fivemembered ring. The present work reports our findings on the π-electron configuration of cyclopentaderivatives of benzenoid hydrocarbons, possessing two or more five-membered rings. It is worth noting that some preliminary researches along these lines were undertaken as early as in the 1970s. 19 Some background theories required to deduce the main results herein are reviewed prior to discussion on the findings in the present study. Theoretical Chemical graph theory The chemical graph theory is well documented in literature. 0,1 In this section we mention some results needed for the subsequent considerations. Let G be a (molecular) graph, possessing n vertices. Let ( G ) be the adjacency matrix of G, and let λ1 λ λn be the eigenvalues of ( G ), referred to as the eigenvalues of the graph G. s is well known, 0- these eigenvalues are closely related to the Hückel molecular orbital (HMO) energy levels of the underlying conjugated π-electron system. Let n +, n, and n 0 be the number of eigenvalues of G that are, respectively, positive, negative, and equal to zero. In HMO theory, n +, n, and n 0 are the number of bonding, antibonding and non-bonding MOs. The characteristic polynomial of the graph G is defined as φ( G, λ) = det[ λ I ( G)] where I is the unit matrix of order n. Then, φ ( G,0) = det[ ( G)] = ( 1) det ( G). On the other hand, det ( G) = λ1 λ λ3 λn. Let G be a graph, and e its edge connecting the vertices u and v. Let Z1, Z,, Zt be the cycles of G in which the edge e is contained. Then the characteristic polynomial of G satisfies the following recurrence relationship, 3-5 φ( G, λ) = φ( G e, λ) φ( G u v, λ) φ( G Z, λ) n t i= 1 i (1) graph is said to be bipartite if its vertices can be coloured by two colours (say, black and white), so that the colour of adjacent vertices is always different.
2 854 INDIN J CHEM, SEC, JULY 010 In Fig. 1 are depicted two bipartite graphs and the colouring of their vertices indicated. graph is bipartite if and only if it does not possess odd-membered cycles. Therefore, the molecular graphs of benzenoid hydrocarbons are bipartite, but those of their cyclopenta-derivatives are not. For bipartite graphs, according to the pairing theorem, λi = λ n + 1 i holds for all i = 1,,, n. Thus, for bipartite graphs, n + = n -. Let G be a bipartite graph, and let n b of its vertices be coloured black and n w coloured white, nb + nw = n. Then CE = CE( G) = nb nw is called the colour excess of G. For example, for the graphs depicted in Fig. 1, CE( G 1) = 0 and CE( G ) =. It is known that n 0 CE, implying that if the colour excess is non-zero, then there exists at least one zero eigenvalue. Then the product of all eigenvalues is equal to zero. Consequently, we arrive at a later used result, Lemma 1: If a bipartite graph G has non-zero colour excess, then det ( G ) = 0 and φ ( G,0) = 0. Topological theory of benzenoid hydrocarbons The fundamentals of the theory of benzenoid molecules is detailed in literature. Here we re-state some of their properties needed for the subsequent considerations. We denote by K( G ) the number of Kekulè structures of the molecule whose molecular graph is G. With respect to K, benzenoid systems are classified as Kekulèan (with K > 0 ) and non-kekulèan (with K = 0 ). It is easy to see that the colour excess of Kekulèan benzenoids is zero and vice versa a benzenoid system with colour excess greater than zero must be non-kekulèan. For instance, one of the benzenoid systems depicted in Fig. 1 has zero colour excess and is Kekulèan, K( G 1) = 9, whereas the other has non-zero colour excess and is non-kekulèan, K(G ) = 0. The Dewar-Longuet-Higgins formula,6 establishes an algebraic relation between the eigenvalues of a benzenoid system and its Kekulè structure count, n/ det ( G) = ( 1) K( G) which, combined with the pairing theorem, yields K( G) = λ λ λn. 1 / Fig. 1 Two benzenoid systems with coloured vertices. In the theory of benzenoid systems it is customary that the peaks are coloured white and the valleys black. The graph G 1 has 1 black and 1 white vertices, and therefore its colour excess is zero. The graph G has 13 black and 11 white vertices and its colour excess is. For what follows we need a special case of the above results, namely, Lemma : If G is a Kekulèan benzenoid system, then the number n of its vertices is even, n+ = n = n /, and n 0 = 0. In other words, within the HMO approximation, a Kekulèan benzenoid system has equal number of bonding and antibonding MOs and has no non-bonding MO. Results and Discussion Monocyclopenta-derivatives of benzenoid hydrocarbons Theorem 3: If B is any Kekulèan benzenoid system, and H is its any monocyclopenta-derivative, then n+ ( H ) = n ( H ) and n0 ( H ) = 0, i.e., H has equal number of bonding and antibonding MOs, and has no non-bonding MO. Proof: Let B be a Kekulèan benzenoid system, and let its vertices be coloured in the above described manner (for example see Fig. ). Then the fivemembered ring of H is attached to B either via two white or via two black vertices. Without loss of generality we assume that the five-membered ring is attached to two white vertices p and q (cf. Fig. ). The two newly added vertices of H may be labelled as u and v, and the edge between them by e (cf. Fig. ). Further, let H(k) be the graph in which the edges of H, connecting the vertices p and u, and q and v have weights k (cf. Fig. ). This means that for k = 1 the graph H(k) coincides with H, whereas for k = 0 the graph H(k) consists of two disjoint components: B and a two-vertex path (cf. Fig. ). In view of the structure of H(0), the set of its eigenvalues is the union of the eigenvalues of B and { 1, + 1}. Therefore, n ( H (0)) = n ( H (0)) and n 0 ( H (0) = 0.
3 GUTMN et al.: π -ELECTRON CONFIGURTION OF CYCLOPENT-DERIVTIVES OF BENZENOID HYDROCRBONS 855 Fig. Kekulèan benzenoid system B, its cyclopenta-derivative H and the auxiliary graphs used for proving Theorem 3. For details see text. pplying Eq. (1) to H(k) we obtain, φ( H ( k), λ) = φ( H ( k) e, λ) φ( H ( k) u v, λ) t k φ( H ( k) Zi, λ) i= 1 () We have now to observe that all the graphs occurring on the right-hand side of Eq. () are bipartite. In particular, (a) H ( k) e is the graph obtained from B by attaching pendent vertices to its vertices p and q. Since both p and q are white, the two new vertices must be coloured black. Consequencely, the colour excess of H ( k) e is equal to two. Then, according to Lemma 1, φ( H ( k) e,0) = 0. (b) H ( k) u v is just the benzenoid system B. (c) The subgraph H ( k) Zi may be viewed as being obtained by deleting from B a path starting at vertex p and ending at vertex q. Since p and q have equal colours, this path contains an odd number of vertices. Therefore H ( k) Zi is a bipartite graph with odd number of vertices. Then, according to Lemma 1, φ( H ( k) Z i,0) = 0 for all i = 1,,, t. Bearing the above in mind, setting λ = 0 into Eq. (), and recalling Lemma, we get n/ φ( H( k),0) = φ( B,0) = det ( B) = ( 1) K( B) 0 Thus, the product of the eigenvalues of H(k) is independent of k, and is non-zero. This means that if the parameter k monotonically increases from k = 0 to k = 1, no eigenvalue of H(k) will change sign. Since n ( H (0)) = n ( H (0)) and n 0 ( H (0) = 0 it must be n+ ( H ( k)) = n ( H ( k)) and n0 ( H ( k ) = 0 for any value of k. Then, for k = 1 we arrive at the claim of Theorem 3. Dicyclopenta-derivatives of benzenoid hydrocarbons In the case of dicyclopenta-derivatives of benzenoid hydrocarbons we must distinguish between two cases. We refer to them as syn and anti. Two cyclopentadiene fragments are in syn position, if both five-membered rings are attached to vertices of the same colour of the parent benzenoid system. Two cyclopentadiene fragments are in anti position, if the two five-membered rings are attached to vertices of different colour of the parent benzenoid system. For an illustrative example see Fig. 3. Theorem 4: If B is any Kekulèan benzenoid system, and S is its any syn-dicyclopenta-derivative,
4 856 INDIN J CHEM, SEC, JULY 010 Fig. 4 Vertices, edges and five-membered rings of S. Notation used in the proofs of Theorems 4 and 5. Fig. 3 G 3 is a syn-dicyclopenta- and G 4 an anti-dicyclopentaderivative of a benzenoid system G 5. The colouring of vertices of G 5 is indicated. then n+ ( S) = n ( S) and n0 ( S ) = 0, i. e., S has equal number of bonding and antibonding MOs, and has no non-bonding MO. Proof is similar to that of Theorem 3. The vertices, edges, and five-membered rings of S are labelled as indicated in Fig. 4. Let S(k) be obtained by assigning a weight k to the edges pu, vq, rx, and sy of the graph S. Then a twofold application of Eq. (1) yields, 4 φ( S( k), λ) = α β k γ k δ where α = φ( S( k) e f, λ) φ( S( k) e x y, λ) k φ( S( k) e ZB, λ) B k φ( S( k) u v ZB, λ) B (3) β = φ( S( k) u v f, λ) φ( S( k) u v x y, λ) γ (4) φ ( S( k) Z f, λ) φ( S( k) Z x y, λ) = k φ( S( k) Z ZB, λ) B (5) δ = φ( S( k) Z, λ) (6) In the above formulas, and B indicate summation over cycles that, respectively embrace the ring but not the ring B, embrace the ring B but not the ring, and embrace both rings and B. In the case syn-sicyclopenta-derivatives, the situation is much simplified since all subgraphs occurring in Eqs. (3)-(6) are bipartite and, with the exception of S( k) u v x y, have colour excess greater than zero. In particular, CE( S( k) e f ) = 4, CE( S( k) e x y) =, CE( S( k) e Z B ) = 3, CE( S( k) u v f ) =. CE( S( k) u v Z B ) = 1, CE( S( k) Z f ) = 3, CE( S( k) Z x y) = 1, CE( S( k) Z ZB ) =, and CE( S( k) Z ) =. Therefore, settingλ = 0, noting that S( k) u v x y B, and by applying Lemmas 1 and, we get φ n/ ( S( k),0) = φ( B,0) = det ( B) = ( 1) K( B) 0 This implies that the product of the eigenvalues of S(k) is independent of k, and is non-zero. Theorem 4 follows now in a fully analogous manner as Theorem 3. Theorem 4 remains valid also if there are more than five-membered rings, provided all of them are in syn-constellation. The case of anti-dicyclopenta-derivatives is significantly more complicated, and for such polycyclic conjugated systems the equality of number of bonding and antibonding MOs may be violated. 15,16,19 From a mathematical point of view, the difficulties arise because several bipartite subgraphs occurring in Eqs. (3)-(6) have now zero colour excess. Theorem 5: Let B be a Kekulèan benzenoid system, and R be its any anti-dicyclopenta-derivative. Then in
5 GUTMN et al.: π -ELECTRON CONFIGURTION OF CYCLOPENT-DERIVTIVES OF BENZENOID HYDROCRBONS 857 the notation specified above and in Fig. 4, n ( R) > n ( R) will hold provided the inequality K( B) + K( B p q r s) < B 4 K( R Z Z ) + K( R Z ) B (7) is satisfied. In other words, if the above inequality holds, then R has more bonding than antibonding MOs. The proof for the above will be only sketched. φ( R, λ ) is expanded in the same way as before (cf. Eqs (3)-(6)) and from φ ( R,0) the terms with nonzero colour excess are eliminated. Then, to the remaining terms the Dewar-Longuet-Higgins formula is applied, noting that the number of vertices of R is four greater than the number of vertices of B. Finally, it can be proven 7 that all pairs of cycles Z, Z for which K( R Z ZB ) 0 have a total number of vertices divisible by four, and this also holds for the cycles Z for which K( R Z ) 0. This leads to the conclusion that if the number n of vertices of B is divisible by 4, then the product of the eigenvalues of R is equal to K( B) + K( B p q r s) 4 K( R Z ZB ) K( R Z ) B whereas the product of eigenvalues of B is equal to K( B ) > 0. If n is even, but not divisible by four, then the product of the eigenvalues of R is equal to B Prolate-rectangle-shaped benzenoid hydrocarbons: case study It is evident that in order that inequality (7) be satisfied, K(B) should be small. One class of Kekulèan benzenoid systems with relatively small number of Kekulè structure are the so-called prolate rectangles. 8 Their general structure, denoted by R( n, m), n, m 1, is depicted in Fig. 5. The anti-dicyclopenta-derivatives of prolate rectangles R(n,m) with m > 1 have numerous pairs of disjoint odd-membered cycles Z, Z B, see Fig. 6. Furthermore, some of the subgraphs obtained by deleting these cycles have large Kekulè structure counts, causing the value of 4 K( R Z ZB ) B in (7) to significantly exceed the value of K( B) + K( B p q r s). For instance, in the case of the anti-dicyclopenta-derivative of R(4,) depicted in Fig. 6, K( B) + K( B p q r s) = = 81 which is much smaller than 4 K( R Z ZB ) = B 4( =... > 3000 fully analogous case is encountered at all antidicyclopenta-derivatives of prolate rectangles R(n,m) with m > 1, enabling one to formulate: Rule 6: ll anti-dicyclopenta-derivatives of prolate rectangles R(n,m) with m > 1 have n+ > n, that is, have more bonding than antibonding MOs. K R Z ZB + K R Z B 4 ( ) ( ) K( B) K( B p q r s) whereas the product of eigenvalues of B is equal to K( B) < 0. In both cases, the validity of Eq. (7) implies that the parity of the number of positive eigenvalues of B and R is different. Since n+ ( B) = n ( B), this implies n+ ( R) n ( R). Knowing that the presence of five-membered rings can only increase the number of bonding MOs, 19 we arrive at n ( R) > n ( R). Fig. 5 The prolate rectangle R(n,m), possessing mn + (m 1)(n 1) hexagons and (n+1) m Kekulè structures. 8
6 858 INDIN J CHEM, SEC, JULY 010 Fig. 6 Some, among the more than one hundred pairs, of the disjoint odd-membered cycles Z, Z B of an anti-dicyclopenta-derivative of R(4,). The Kekulè structure counts of the cycle-deleted subgraphs, pertaining to diagrams 1,,, 8, are 9, 14, 14, 14, 9, 0, 1 and 0, respectively. (For completeness, recall that all syn-dicyclopentaderivatives of prolate rectangles R(n,m) with m > 1 have n+ = n, that is, have equal number of bonding and antibonding MOs.) Corroborated by our extensive numerical calculations, the above rule can be extended to arbitrary cyclopenta-derivatived of R(n,m): Rule 7: cyclopenta-derivative of prolate rectangles R(n,m) with m > 1 has n+ > n, and thus more bonding than antibonding MOs, if and only if it possesses at least one pair of five-membered rings in anti-constellation. Otherwise, n = n. The case of prolate rectangles R(n,m) for m = 1, namely the case of linear polyacenes, is exceptional (cf. Fig. 7). If m = 1, then the arguments leading to Rule 6 do not apply, and also Rule 6 does not generally hold. By numerical testing we found that there exist anti-dicyclopenta-derivatives of R( n,1) Ln, for which n+ = n, i. e., which have equal number of bonding and antibonding MOs. The smallest such molecules are depicted in Fig. 7. For larger values of n, such violations of Rule 6 are more numerous. For instance, n+ = n holds for L 1 (1,8), L 1 (1,9), L 1 (1,10), L 1 (1,11), and L 1 (,10). Evidently, the condition for the equality of the numbers of bonding and antibonding MOs is that the two cyclopentadiene fragments are sufficiently far from each other. The numerical data obtained for n-values up to 15 indicate that n+ = n holds for Ln (1, t ) whenever t ( n 1) / 3 + ( n 1) / 3 + 1, which remains a challenge for to be mathematically verified for the general case.
7 GUTMN et al.: π -ELECTRON CONFIGURTION OF CYCLOPENT-DERIVTIVES OF BENZENOID HYDROCRBONS 859 Fig. 7 The linear polyacene Ln R( n,1) and the labelling of its anti-dicyclopenta-derivatives. L 4 (1,3), L 5 (1,4), L 6 (1,4), and L 6 (1,5), and are the smallest such molecules having equal number of bonding and antibonding MO, thus violating Rule 6 for m = 1. Conclusions The present study was aimed at revealing the structural conditions required by the cyclopentaderivatives of benzenoid hydrocarbons to have equal number of bonding and antibonding Hückel molecular orbitals. It has been recognized 19,9-31 that this is a necessary condition that the respective molecule has a stable π-electron configuration, and thus be chemically stable. This was also confirmed by our recent studies on dicyclopenta-species. It is hoped that the general results reported in this paper will shed more light on this problem, and in addition, show how the mathematical apparatus of chemical graph theory could be used for the treatment of problems of this kind. cknowledgement The authors thank the support by the Serbian Ministry of Science, through grant no G, and project "Graph Theory and Mathematical Programming with pplications to Chemistry and Engineering". References 1 Clar E, The romatic Sextet (Wiley, London) 197. Gutman I & Cyvin S J, Introduction to the Theory of Benzenoid Hydrocarbons (Springer-Verlag, Berlin) dvances in the Theory of Benzenoid Hydrocarbons, edited by I Gutman & S J Cyvin, (Springer-Verlag, Berlin) dvances in the Theory of Benzenoid Hydrocarbons II, edited by I Gutman (Springer-Verlag, Berlin) Randić M, Chem Rev 103 (003) Gutman I, & Đurđević J, MTCH Commun Math Comput Chem, 60 (008) Gutman I, Đurđević J, & Balaban T, Polyc rom Comp, 9 (009) 3. 8 Đurđević J, Gutman I, Terzić J & Balaban T, Polyc rom, Comp, 9 (009) Balaban T, Đurđević J & Gutman I, Polyc rom Comp, 9 (009) Gutman I, Đurđević J, Radenković, S & Burmudžija, Indian J Chem, 48 (009) Gutman I, Jeremić S & Petrović V, Indian J Chem, 48 (009) Marković, S, Stanković S, Radenković S &. Gutman I, Monat Chem, 140 (009) Đurđević J, Radenković, S, Gutman I & Marković S, Monat Chem, 140 (009) Balaban T, Dickens T K, Gutman I & Mallion R B, Croat Chem cta, 8 (010), (In press). 15 Gutman I & Furtula B, Polyc rom Comp, 8 (008) Gutman I, Đurđević J, Furtula B & Milivojević B, Indian J Chem, 47 (008) Marković S, Stanković S, Radenković S & Gutman I, J Chem Inf Model, 48 (008) Stanković S, Marković S, Radenković S & Gutman I, J Mol Model, 15 (009) 953.
8 860 INDIN J CHEM, SEC, JULY Gutman I, Trinajstić N & Živković T, Tetrahedron, 9 (1973) Trinajstić N, Chemical Graph Theory, (CRC Press, Boca Raton) Gutman I & Polansky O E, Mathematical Concepts in Organic Chemistry, (Springer-Verlag, Berlin) Dias J R, Molecular Orbital Calculations Using Chemical Graph Theory, (Springer-Verlag, Berlin) Heilbronner E, Helv Chim cta, 36 (1953) Schwenk J, in Graphs and Combinatorics, edited by R Bari & F Harary, (Springer-Verlag, Berlin) 1974, pp Gutman I & Polansky O E, Theor Chim cta, 60 (1981) Dewar M J S & Longuet-Higgins H C, Proc Royal Soc London, 14 (195) Gutman I & Đurđević J, MTCH Commun Math Comput Chem, 65 (011) Cyvin S J & Gutman I, Kekulè Structure in Benzenoid Hydrocarbons, (Springer-Verlag, Berlin) 1988, pp Streitwieser, Molecular Orbital Theory for Organic Chemists, (Wiley, New York) Dewar M J S, The Molecular Orbital Theory of Organic Chemistry, (McGraw-Hill, New York) Dewar M J S & Dougherty R C, The PMO Theory of Organic Chemistry, (Plenum Press, New York) 1975.
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