Mobius Molecules and Graphs*

Transcription

1 CROATICA CHEMICA ACTA CCACAA 47 (2) (1975) YU ISSN :516 CCA-85 8 Original Scientific Pa,per Mobius Molecules and Graphs* A. Graovac and N. Trinajstic** Th e Rugjer Boskovic I nstitute, P.. B. 116, 411 Zagreb, Croa tia, Yugoslavia Received February 17, 1975 Mobius molecules and graphs are discussed. A brief discu ssion of the generalized graph s is also given. The extended Sachs formula for Mobius graphs (and generalized graphs) is.r eported and some examples discussed. The parity of th e algebraic structure of Mobius molecules are defiined. In recent years the use of the Huckel-Mobius concept 1 is becoming increasingly popular 2-4 for studying molecules and reactions. Mobius systems are defined 5 as cyclic arrays of orbitals in which there is one sign inversion, or more generally, in which there is an odd number of sign inversi ons resulting from the nega tive overlaps between the adjacent orbitals of different sign. Mobius cyclic polyenes (called Mobius annulenes) may appear in solution as possible twisted conformational forms of.higher annulenes 6 (CH) 11, n 2. Mobius-like structures may also appear as tran sition states in the electrocyclic closures of linear polyenes. For example, t h e conrotatory closu re of butadiene and the disrotatory closure of hexatriene should prefer the Mobius transition state 1, 7. On the other hand, annulenes are composed of orbital arrays in which there is no sign inversion, or more generally, in which there is an even number of sign inversions (positive-negative Overlaps) among the adjacent 2pz-orbitals. These molecules are called Ruckel molecules because they obey the Ruckel rule 8 w hich says t hat a planar cyclic polyenes with 4m + 2 l1-electrons (where m is an integer) should be stable and aromatic compounds. However, the Ruckel systems with 4 m l1-electrons are predicted to be very reactive and antiaromatic species. The chemistry of annulenes is in accordance with this rule Fo r Mobius system s it is predicted 5 11 that the systems with 4m ;c-electrons would exhibit aromatic stability whereas those with 4m + 2 l1-electrons wou)d be antiaromatic. Therefore, the Mobius systems would display anti-ruckel properties and are indeed often called 12»anti-Ruckel systems«. The aim of the present work is to show that graph-theoretical oonsiderations can b e extended to Mobius systems and, consequently, that a number of important pieces of information about these systems can be extracted from simple graphical (graph-theoretical) rules. Graph theory is widely applied to Ruckel systems 13 and a number of interesting results are o btained 14-2:i. Ruckel > * Reported in p art at the Meeting of Croatian Chemists, Febr uary 1975, Zagreb, Croatia, Yugoslavia. ** Also at the Chemistry Department, Faculty of Science and Mathematics, University of Zagreb.

2 96 A. GRAOVAC AND N. TRINAJSTIC systems can be represented by Ruckel graphs which correspond to n-network of a conjugated molecule. Thus, the correspondence between the Ruckel molecules and Ruckel graphs is one-to-one. Since the Ruckel Hamiltonian can be written in matrix form as a unique function of the Ruckel graph 24, the equivalence between the Ruckel theory and the graph spectral theory is established In order to extend the graph-theoretical approach to Mobius systems we need to define a new type of graphs which we will call Mobius graphs*. The weight of edges 13 in these graphs is either + 1 or - 1 depending on whether two 2p,-atomic orbitals in a Mobius molecule are in the positive-positive (+ 1) or in the positive-negative (- 1).overlap relationships, respectively. In our previous works 13 we have studied only graphs with the edge weight of + 1. As an example, we give below graphs which correspond to Ruckel (G 1 ) and Mobius (GJ cyclobutadienes, respectively. tj tl The location of the connectivity - 1 between two vertices in the Mobius graphs of annulene is arbitrary. However, the important information is given that one (odd) sign inversion exists in the structure called Mobius cyclobutadiene. Note that the Mobius graphs discussed here are a spe6al case of more general graphs with an arbitrary number of - 1 edges which we call the generalized graphs**. Generalized graphs occur, for example, when the graph theory is applied to systems treating the relationships between the people, thus, the + 1 edge would symbolized the attmction between two persons, O indifference, and the - 1 edge repulsion. In another words, in the case of generalized graphs there are two distinctive, binary relations 1 '\ but of oppdsite meaning between the pairs of elements of a system. The system is then fully defined when the binary relations between all pairs of elements are established. Mobius graphs can be also described by the (vertex) adjacency matrix 28 A (G) of a special type, defined as follows: 1-1 if there is a positive edge between the adjacent vertices r and s if r = s, or if there is no edge of any kind between the vertices r and s if there is.a negative edge between the adjacent vertices r and s * There should be no confusion with the Mi:ibius ladder graphs, which are derived from a circuit graph containing an even number of points by adding new edges joining each pair of opposite vertices (See, for example, N. Biggs, Algebraic Graph Theory, Cambridge University Press 1974, p. 2). ** There are some proposals to label these graphs differently such as the extended graphs 27, and consequently to call the theory the Extended Graph Theory.

3 MbBIUS MOLECULES AND GRAPHS 97 The foll-owing matrices assigned to Ruckel and Mobius cyclobutadiene graphs, respectively, l may ; be considered as their vertex adjacency matrices: A (G,) ~ j A (G2) = ~ _ ~ j l Now, we wish to demostrate how the structure of the Mobius graph is related to the various coefficients appearing in the characteristic polynomial P (G, x) of the vertex adjacency matrix: N P (G,x) = det I x 1 - A I = ~ a,, x N - n (1) of such a graph with N vertices, where a,, are the coefficients of the characteristic polynomial, and to show also that the construction of the characteristic polynomial appropriate to a Mobius graph is, once again 29, a purely combinatorial problem. Evaluation of the coefficients a n of the characteristic polynomial corresponding to a Mobius graph may be carried out in the same way as described by Graovac et al. 3 for Ruckel graphs using the Sachs formula 31 However, in order to do that, we have to extend the original Sachs formula to cover both Ruckel and Mobius graphs. We give below the extended Sachs formula: ao = 1 (per definitionem) n = O a n = ~ (-)C(s) + pr(s) 2r(s) for 1 :'( n :'( N SGS (2) where the symbols have the following meaning: s denotes a Sachs graph 3, Sn is a set of all Sachs graphs (with n vertices) of a graph G, c(s) is the total number of components of a Sachs graphs s, r(s) is the number of rings of a Sachs graph s, and finally Pr(sl is the number of - 1 edges in the rings of Sachs graph s. The summation in (2) is over all elements of the set Sn. We shall illustrate the application of the extended Sachs formula to Mobius cyclobutadiene. First all Sachs graphs of G 2 should be constructed: s, = ~ Then, we use formula (2) in order to evaluate a,,'s: a = 1 a 1 = a2 = 4 (-)1 + 2 = -4 a 3 = U4 = 2 (-) (-) = 4

4 98 A. GRAOVAC AND N. TRINAJSTIC Finally, the fo11owing characteristic polynominal is obtained: P (G 2, x) = x 4-4x 2 + 4; { x = -..;2, -..;2, - -../~ -- -V2} It can be compared with the characteristic polynomial belonging to Ruckel cydobutadiene: P (G 1, x) = x 4-4x 2 ; { x = 2,,, - 2 } Obviously polynomials P (G 1, x),and P (G 2, x) differ only in the value of the an (N = 4) coefficient. Similar result can be also obtained for Ruckel (G ) 3 and Mobius (G 4 ) benzenes: P (G 3, x) = x 6-6x 4 + 9x 2-4; { x = 2, 1, 1, - 1, -- l, - 2 } -1 P (G., x) = x 6-6x 4 + 9x 2 ; { x = -y1 3, -y1 3,,, - -y1 3, - -y1 3 } A general characteristic feature of all monocyclic Ruckel and Mobius systems is that they differ in the value of the ON coefficinet only. (This is not surprising because monocyclic systems contain the cycles only in SN set of Sachs graphs). This coefficient provides information about the chemical stability, namely, the non-vanishing an value indicates 3 the closed-shell stability and aromaticity, whereas an = is characteristic 16, 3 of monocyclic systems with open-shell reactivity and antiaromaticity. This can then be used for a very simple classification scheme.of all 4m (mod 4) 31 a and 4m -- (mod 4) 31 a generalized monocyclic systems: an = - 4; e.g. an ~ an = Ruckel (4m + 2)-systems Mobius (4m)-systems Ruckel (4m)-systems Mobius (4m + 2)-systems Another property of coefficient an is also interesting; an is related 3, 32 to the, number of even (K-, parity= + 1) and odd (K-, parity= -1) Kekule structures 32 (K = K + + K-) of Ruckel molecules: The parity of a Kekule structure can be obtained using the following rule3,33,34: two Kekule structures have opposite (identical) parities if the superposition of the corresponding Kekule graphs contains odd (ev en) number of (4m)-membered rings. The Eq. (3) could be applied to Mobius systems. However, it is more appropriate to call the structures of Mobius systems related to the Kekule structures of Ruckel systems algebraic structures (AS), because they (3)

5 MOBIUS MOLECULES AND GRAPHS 99 are related to permutation matrices in exactly the same way as Kekule structures33,35. Thus, Kekule structures of Ruckel molecules are just a special case of the algebraic structures which thus cover both Ruckel and Mobius systems. Therefore, eq. (3) can be more appropriately rewritten as follows : an = (-)N/2 1(ASt - (4) (ASr 12 where (AS) + and (AS) - are the algebraic structures of even and odd parity. Now, we need define the parity of algebraic structures. We denote graphs corresponding to algebraic structures by h a, h b,... Let the superposition gra ph of h a and hb be labelled as Sab Let also the number of Ruckel-type ring components of the size (mod 4) in the superposition graph Sab be H(Sab) and the number of Mobius-type ring components of size 2 (mod 4) in Sab be M (Sab). Furthermore, let p = + 1 for even and p = - 1 for odd algebraic structure. Then, the equation (5) completely determines the parity of algebraic structures. Namely, two algebraic structures ha and hb are of the same parity if, and only if, the sum of H (Sab) + + M (Sab) is even. The use of formulae (4) and (5) is illustrated for Ruckel and Mobius cyclobutadienes and benzenes, respectively, in Table I. The total n-electron energies (E = Na + N/2 ~ x ; fj) of Ruckel (E ~i = 4a + 4fJ) i~l and Mobius (E;vi = 4a + 4 v2// ; fj' = fj cos n!n; N = 4) cydobutadienes are equal (E;f- = E~ ). However, Ruckel cyclobutadiene has a reactive open-shell configuration whereas Mobius cyclobutadiene has a closed-shell configuration. This is a general characteristics of 4m systems. The situation is different for 4m + 2 systems. Thus, Ruckel benzene has E-;:6 + 8fJ, whereas Mobius benzene has E!1 = 6a + 6/3 (E~'l = v3 j3'; fj' = j3 cos n!n; N = 6); the difference being 2fJ in av.o ur of Ruckel benzene. Ruckel 4m + 2 systems have always larger value's of E" than the corresponding Mobius systems. However, this difference for a very large system is negligible (fj' --+ fj for N--+ oo). We have also considered extension of these studies to some bicyclic systems which could be classified as Huckel-Mobius (G:;) and Mobius-Mobius (G 6 ) systems, respectively, (where the individual Mobius part has only one = -1(--~-)-1 --~~- phase dislocation, i.e. - 1 edge).* Huckel-Mobius graphs (G ) considered here consist of two rings which have only two vertices and an edge in common. One ring is a Ruckel-type cycle (H) and the other a Mobius-type cy cle (M). * In general, the /?-parameter of the theory is different in Hiickel (/?) and l\/iobius (/?') parts of Hiickel-Mobius bycyclic molecules. Since we are here interested only in. giving a qualitative study, the difference between p and P' is neglected. However, a very general Sachs formula 36 for graphs with edges of different weights could be applied.

6 TABLE I >--' Parity of algebraic structures of Hiickel and Mobius cyclobutadienes and benzenes Super- Molecule Algebraic positio~ of Parity of algebraic a = (-ti I (ASt _ (Asr 12 structure algebraic structures N structues Hi.ickel cyclobutadiiene I [J}I A assumed: + 1 ~~ bemene I I I l lldj { H (S b) 1 :i> _ calculated: -1 M (S:b) ~ '"urned, :i> + 1 ~ { -4 ~ H (Sab) = calculated: + 1 M (S ) = ~ ili ~ ~~~~~~tadiene ~ \ (I ii -1 lu assumed: +_1 ~ I -1-4 ~ F9 J. { H (Sabl = o 1=J -1 I calculated. + 1 M (Sabl = Mi::ibius ~ bemene _ I I '"urned' + 1 I. { H (Sab) = calculated. - 1 M (S ) _ 1 ili - 1 o I

7 MdBIUS MOLECULES AND GRAPHS 11 Mi:ibius-Mi:ibius graphs (G ) are very closely related to Hiickel-Mi:ibius graphs except that in the former graphs both rings are of Mi:ibius-type. We will consider, as an example, Hiickel-Hiickel {G 7 ), Hiickel-Mi:ibius (Gs), and Mi:ibius -Mi:ibius {G 9 ) bicyclohexatrienes (butylenes), respectively. Et] ~ G-i First, we give below all Sachs graphs of G,: =\(x; ).(- ).( <>-<>). ( x: ).( :r),( o-ouo-o J} 5 l. G 5 Actually, these are formally all possible Sachs graphs not only of G 7, but G 8 and G 9 However, the corresponding characteristic polynomi als would differ iii those values of a,, coefficients for which the cycles a1ppear in the Sachs graphs. In our case these would be a, and a coefficients because S, and s.. Sachs' graphs of Gs and G 9 contain rings with - 1 edges. Now the coefficients a" of the characteristic polynomials P (G 7, x), P (Gs, x), and P (G, x ) can be evaluated using the extended Sachs formula (2): a = 1 a 1 = a2= 7 (- )1 +2=-7 U 3 = a., (G,) = 11 (-) (-) = 7 a., (G ) 8 = 11 (-) (-)l +O 21 + (-) = 11 a, (G 9 ) = 11 (- ) (-) = 15 a,,= a.. (G ) 7 = 3 (-) (-) (-) = -1 a (G ) 8 = 3 (-)l +O 2 + (- ) (- ) (-) = -1 a,, (G ) 9 = 3 (-) (-) (-) = _ g Finally, the following characteristic polynomials are obtained: P (G 7, x) = x 6-7x 4 + 7x 2-1; P (Gs, x) = x 6-7x 4 + llx 2-1; P (Gn, x) = x 6-7x x 2-9; { x = 2.41, 1.,.41, -.41, - 1., } { x = 2.17, 1,48,.31, -.31, , } {x = 1.73, 1.73, 1., -1., -1.73, -1.73} The value of a.n coefficients can be also obtained by using formulae (3) and (4). This is reported in Table II.

8 ... tv TABLE II The calculation of an for G 7, G 8, and G 9 (see text) Algebraic Superposition of I Paritiy of algebraic a = (-)Nf' 1 (ASj+ _ (Asr \2 Molecule structure algebraic structure structures N : Hiickel-Hiickel C[J assumed: + 1 bicyclohexa- - i> triene H (Sahl = (G 7 ) ri-r=1 rr=r==v calculated: + 1 M (S ) = -1 ~ l!_l=.j bbdj ab i> H (Sahl = 1 ~ DO calculated: -1.. i> ' M (Sahl = n I CT] i> ~iickel-mobius I I JI - 1 assumed: + 1 z tj b1cyclohexa z triene H (Sahl = (Gs) [D -1 le:lj -1 calculated: -1 M (Sb)= 1-1 ~ a Z H (Sahl = ~ m -1-1 calculated: + 1 [fl I L.lLJ M (Sahl = :::1 () Mobius-Mobius -1 C[] -1 assumed: + 1 bicyclohexa- - triene. H (Sahl = (G9) -1 II I I -1-1 EC] -1 calculated:+ 1 M(Sabl = - 9

9 MOBIUS M OLECULES AND GRAPHS 13 an values from Table II are, of course, identical with those appearing in P (G 7, x), P (Gs, x ), and P (G, x ), respectively. A final point we wish to discuss is this: under which condihons our procedure is invariant with respect to the position of a - 1 edge in the Mobius-. type graphs described here. For monocyclic graphs, because of the high symmetry of annulenes, the invariance on the position of the - 1 edge is easily seen. However, this is not so evident for Huckel-Mobius (G 5 ) and Mobius-Mobius (G 6 ) graphs. Let us first consider the bicyclic graphs G 1 with - 1 common edge. It becomes apparent from the insipection of Sachs graphs containing rings that G 1 belongs to the class of Mobius-Mobius bicyclic.graphs. Thus, for example, Mobius-Mobius graph G 11 has the characteristic polynomial, P (G;i, x) identical with the one of G, P (G, x). Similarly, it could be easily shown by inspection of the S, and Sc Sachs graphs that the characteristic polynomials of Huckel-Mobius graph G 12, P (G 12, x ), and Mobius-Mobius graph G 1 3, P (Gm, x) are identical with P (Gs, x) and P (G 8, x ) respectively. _, ITJ Therefore, it follows from the inspection of G 11, Gw and G 1 3 that two Hi.ickel -Mobius or Mobius-Mobius graphs are invariant with respect to the position of the -1 edge if the Sachs graphs containing rings are in both graphs identical. Acknowledgements. We thank Dr. D. Cvetkovic': (Belgrade), Dr. I. Gutman (Visoko), and Professor M. Randie (Medford) for helpful correspondence. P art of this work is accomplished during a stay of one of us (N. T.) at the Rechnenzentrum der Universitat Dilsseldorf. N. T. thanks Dr. J. V. Knop, th e Head of the Rechenzentrum,for his kindness in making available facilities at the Recehnzetrum and for helpful discussions. REFERENCES 1. H. E. Zimmermann, J. Amer. Chem. Soc.!18!1966) 1564, H. E. Z i m m e rm a n n, Accounts Chem. Res. 4 (1971) K. - W. Sh e n, J. Chem. Educ. 5 (1973) H. E. Zimmermann, and R. D. Litt 1 e, J. Amer. Chem. Soc. 96 (1974) 5143; W. J. Hehr e, J. Amer. Chem. Soc. 95 (1973) 587; G. A. 1 a h, J. S. S t a r a 1, and G. Li a n g, ibid. 96 (1974) E. He i 1 b r on n er, Tetrahedron L ett. (1964) 1923.

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