Mathematical relations necessary for the calculation of the
previously introduced15 topological resonance energy, TRE, are
derived. TRE, as defined in ref. 15, was rather inconvenient for
numerical calculations. In the present work these computing difficulties
are overcome.
Acyclic characteristic polynomial pac and acyclic spectrum of
a graph are defined and their close relationship to TRE is demonstrated.
Hence, the evaluation of pac is the crucial .step in the computation
of TRE\u27s. A graphical procedure is described and exemplified
for the determination of P"c\u27s both for simple graphs and
for graphs with loops and weighted edges

I. Gutman

M. Milun

N. Trinajstić

English

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CROATICA CHEMICA ACTA CCACAA 48 (2) 87-95 (1976) CCA-919 YU ISSN 0011-1643 5319.19 :516 Original Scientific Paper Graph Theory and Molecular Orbitals. XVIII. On Topolog·ical Resonance Energy* I. Gutman, M. Milun**, and N. Trinajstic The Rugjer Boskovic Institute, P.O.B. 1016, 4100l Zagreb, Croatia, Yugoslavia and** The Pharmaceutical and Chemical Works »Pliva«, 41000 Zagreb, Croatia, Yugoslavia Received November 2, 1975 Mathematical relations necessary for the calculation of the previously introduced15 topological resonance energy, TRE, are derived. TRE, as defined in ref. 15, was rather inconvenient for numerical calculations. In the present work these computing diffi-culties are overcome. Acyclic characteristic polynomial pac and acyclic spectrum of a graph are defined and their close relationship to TRE is demon-strated. Hence, the evaluation of pac is the crucial .step in the com-putation of TRE's. A graphical procedure is described and exem-plified for the determination of P"c's both for simple graphs and for graphs with loops and weighted edges. INTRODUCTION Resonance energy, RE, is defined as RE = E(molecule) - E (reference ) structure (1) where E(molecule) is the total (or total rr-electron) energy of a conjugated system, whe.reas E(~fi~~~:Ji: ) is the analogous quantity for a hypothetical re-fevence structure. Depending on the choice of this reference structure, dif-ferent RE's have been defined1-s. An especially successful concept has proved to be the so called9 Dewar resonance energy, DRE, which is based on an »acyclic polyene-like« reference structure1,3,4,10. In DRE the energy of the reference structure is introduced in the fol-lowing manner. The total energy of certain classes of acyclic systems can be given in a simple additive form10 E ( acyclic ) = ~ n1. E1. polyene j (2) where ni is the number of bonds of a given type contained in the considered molecule and Ei is the corresponding bond energy parameter. Note that eq. (2) holds approximately and the E/s are determined by a least-squares fitting of this .relation to the values of a selected set of acyclic systems and bond types. The basic idea of DRE is that both aromatic stabilization and antiaromatic destabilization have their origin in the presence of cycles in the molecular * Part XVII: I. Gutman and N. Trinajstic, J. Chem. Phys., in press. 88 I. GUTMAN ET AL. rt-network. Therefore, the aim of the method is to find a reference structure which would contain all, and only, acyclic contriibutions to E(molecule). Eq. (2) indicates the choice E (reference) = ~ n. E. structure 1 1 (3) j In fact, the DRE method is defined by eq. (3). The excellent agreement between DRE's and the experimental findings is well estabhshed1,3, 4,9 , io nowaday1s. It is also worth noting that DRE's calculated within the HMO approximafaon are not less accurate than the same quantities obtained by other, more sophisticated SCF MO techniques3,4,11-14 . However, the DRE method is not quite free from conceptual difficulties. Thus, (a) it contains a large number of fitting parameters (E/s) which indicate that the method is to some extent arbitrary; (b) it is not a priori certain that eq. (3) contains all acyclic and no cyclic contributions to E(molecule); (c) the numbers ni can be evaluated only for systems possessing a Kekule structure**, and therefore charged species, radicals etc. are outside the scope of the DRE method. In a previous work15 topological resonance energy, TRE, a novel variant of the DRE has been proposed, which does not suffer from the above listed difficulties. In particular, (a) TRE requires no fitting parameters; (b) it is mathematically defined in such a manner that the energy of the TRE reference structure contains exactly all acyclic and exactly no cyclic contributions to E(molecule); (c) its applicability is not stipulated by the existence of Kekule structures. The starting point in introducing the concept of TRE was a Coulson-type integral formula for the total rt-electron energy16- 18 , Er:, + 00 Jdx E" = (1/2 n) - ., lo.g {[ ~ (-l)n (s) /2 + c (s) X· SE Se . 2r (s) xn (s) ]2 + [ ~ (-l)(n (s) + 1)/ 2 + c (s) 2 r (s) xn (s) ]2} (4) SES0 where Se and S 0 are the sets of all Sachs graphs19 of the molecular graphs with even and odd number of vertices, respectively. The summations in (4) go over all elements of the sets Se and S 0 , and n(s), c(s) .and r(s) are the number of vertices, components and cyclic components, respectively, of the Sachs graph s. Let in addition S = Se U S 0 be the set of all Sachs graphs of the molecular graph. Further details on Sachs graphs are given elsewhere19•20 . Eq. (4) ·indicates that every Sachs graph gives a certain contribution to the value of E". Then, obviously, the contributions of Sachs graphs with r(s) = 0 will represent the acyclic part of the total n-electron energy. Con-sequently, one can define the »acychc polyene-like« reference structure as ** More precisely, n/s are unique only in systems with exactly one Kekule structure. GRAPH THEORY AND MOLECULAR ORBITALS + 00 E (reference) = (1 /2 n) J dx log { ( 2 (-l)n (s) / 2 + c (s). 7t structure x2 SES~C . xn (s)]2 + ( 2 (-l)(n (s) + 1)/2 + c (s) x n (Sl]2} SE S~C 89 (5) where S~c (S~c with even (odd) = s~c u s~c . ) is the set of all Sachs graphs of the molecular graph number of verhces and with r(s) = 0. Let further Sac For graphs becomes representing conjugated hydroca,rbons, s~c Cf> , and eq. (5) + 00 E (reference ) = (1/n J ~ locr { [ 2 (-l)n (s)/ 2 + c (s) • xn (s)] 1t structure x2 ° SE Sac (6) - 00 Eqs. (5) and (6) are the mathematical definitions of TRE15• Although these formulae give a proper insight into the topological origin of aromatiCity (TRE or DRE > 0) and antiaromaticity (TRE or DRE< 0), they are rather im-practical for performing actual numerical calculations. In the present work a graph theoretical technique that enables simple evaluation of TRE's in the general case will be presented. THE ACYCLIC CHARACTERISTIC POLYNOMIAL AND THE ACYCLIC SPECTRUM OF A GRAPH The argument used in the definition of the TRE reference structure (eqs. (5) and (6) can be generalized in ,a simple and straightforward manner. Let F be any quantity which can be calculated from a given set a of Sachs graphs. Symbolically, F = f (s J s E o) Now the »acyclic F« is, by definition Fae = f (s J s E o, r (s) = 0) or in another notation Fae = f (s I s € oac) (7a) (7b) (7c) where aac is the set of all Sachs grnphs s from a with the property r(s) = 0. In the following we will be interested in the »acyclic characteristic poly-nomial«, p ac (G, x), and the »acyclic spectrum« of the graph G. Let the cha-racteristic polynomial P (G, x) = P (G) of the graph G be of the form N P(G) = 2a11xN-n n = O where N is tthe number of vertices of G. Since P (G) fulfills the relation19- 21 P (G) = 2 (- l)c (s) 2T (s) xN -n (s) according to eqs. (7), SES Pac (G, x ) = 2 (- l) C (s) xN - n (s) SE Sac (8) 90 1. GUTMAN ET AL. which will be presented also in the form 'V Pac (G) = ~ aac xN-n n (9) Let the roots of P (G) be x 1 , x 2 , ••• , xN (x; ~ xi if i < j), and these numbers comprise the spectrum of the graph G. Then the roots of pac (G, x), x1ac , x~c , ... , x'J:.? (x~c ~ xjc if i < j) will form the »acyclic spectrum« of the graph G. The chemical meaning of the above introduced notions is evident from the following statement. Proposition 1 If the total n-electron energy fulfills the relation N E" = ~ f xif j = I the .reference structure energy of eq. (5) is N E (reference) = ~ I xqc I " structure J j = I (10) (11) The proof of this proposition is simple. Namely, the Coulson formula (4) is just an integral representation for the sum of the absolute values of the roots of a polynomial16- 18• Eq. (4) describes such a sum for P (G, x), while eq. (5) is the analogous expression for pac (G, x). In the general case, instead of (10), N E"' = ~ gixi j = I where gi is the occupancy number of the j 'th MO. In agreement with the above discussion, eq. (11) can also be generalized as which finally yields N E,, ( ~fi~~f~~:) = ~ gi xjc j=I N TRE = ~ gi <xi-xr) j = I If G is an acyclic graph, and consequently, pac (G, x) = P (G, x) TRE (acyclic ) _ 0 polyene = (12) (13a) (13b) (13c) GRAPH THEORY AND MOLECULAR ORBITALS THE ACYCLIC CHARACTERISTIC POLYNOMIALS FOR MOLECULAR GRAPHS OF CONJUGATED HYDROCARBONS 91 Conjugated hydrocarbons are presented by graphs without loops and weighted edges19,20 • Therefore Sachs graphs from s ac are composed entirely from complete graphs of degree one***. Proposition 2 follows now from eqs. (3) and (9) . Proposition 2 {(- l)n/2 b I aac = n 2 n 0 for n even for n odd where bi = bi (G) is the number of different choices of j nonincident edges from the graph G. (b 0 = 1 by definition.) It seems worth mentioning that for acyclic graphs (cf. eq. (13a)) this proposition has been independently discovered by various researchers21- 24. Let e be an arbitrary edge of the graph G and G-e denote the graph obtained by deletion of this edge from G. Let further G -- (e) denote the graph obtained by deletion of the edge e and its two incident vertices from G. The numbers bi have the following important property. Proposition 3 bi(G) = bi(G-e) + bi--1 (G-(e)) Proof. - Among the bi (G) choices of j nonincident edges from G, there are bi (G-e) such choices which do not contain the edge e and bi-i (G-(e)) choices which contain the edge e. Propositions 2 and 3 give together with eq. (9) the following corrolary. Proposition 4 pac (G, x) = pac (G - e, x) - pac (G- (e), x) (14) This equation has been proved for acyclic graphs (cf. eq. (13b)) many years ago by Heilbronner25 and was recently independently derived24 • Hosoya has obtained the Proposition 4 in a different, but equivalent form22 • Proposition 4 presents a very simple graphical procedure for evaluating the pac's for polycyclic graphs. In fact, eq. (14) enables one to relate p ac of a polycyclic graph with the acyclic characteristic polynomials of acyclic graphs. For these latter gra,phs, on the other hand, eq. (13b) is applicable. As an example let us calculate the p ac of biphenylene. Ln will denote the character-istic polynomial of the chain with n vertices. P"'(C):O) oP''(C=O)-P"'(( 0) ;p''(C::) )-p"'( c ))-p"'( ( 0) *** These graphs contain two vertices joined by an edge. 92 I. GUTMAN ET AL. Now P" ( c ) ) o L, L6 pa' ( c 0 ) oL, p" ( 0 ) 0 Ld L5 - L, I Finally, = x12 -14x10 + 71x 8 -162x6 +164x~ -60x2 +5 The roots of this polynomial can be obtained by standard computer routines. Note that the polynomials L n can be easily evaluated using the well known.25 recurrence relation As another example we consider the p ac of an N-membered cycle, CN, which is, of course, the graph representation of annulenes. Since for an aT-bitrary edge e, Crv-e and CN-(e) are identical with chains of N and N-2 vertices, respectively, It is known24 that sin (n + 1) t L = -----n sin t for x = 2 cos t. Therefore from (15), pac (CN) = 2 cos Nt The acyclic spectrum of CN is now simply obtained from Nt = rr,/2 + jn: (2j-1) Jt xjc = 2 cos --2- N-- j = 1, 2, ... , N (15) GRAPH THEORY AND MOLECULAR ORBITALS and for neutral annulenes { 2 cosec (n/2 N) E ( reference ) " structure - 2 cotg (n/2 N) if N if N even odd · 93 As the third example, we consider the bicyclic system Vn, containing two n-membered cycles joined by an edge e. n n e n n Now, pac (V n) = [Pac (Cn)J2 - (Ln-1)2 = (Ln - L,,_2)2 - (Ln-1)2 On the other hand, the characteristic polynomial of the graph Wn (with 4n-2 vertices)* is according to eq. (14) P (W n) = (L2n-1l 2 - (Ln-1)4 = (Ln-1)2 [(Ln - Ln-2) 2 - (Ln- 1Fl and hence one obtains an amusing relation TRE (V n) = E,, (V n) - E" (W n) + 2 E,, (Ln_1 ) Using eq. (14), pac's for a large number of molecular topologies have been calculated, together with the corresponding acyclic spectra and TRE values. These will be reported elsewhere26 • THE ACYCLIC CHARACTERISTIC POLYNOMIALS FOR GRAPHS OF HETEROCONJUGATED MOLECULES In the graph themetical representation of conjugated molecules, a hetero-atom is presented by a loop of the weight h attached to the vertex corresponding to the heteroatom27•28• The edges rncident on such a vertex have the weight k. Therefore, Propositions 2-4 are no more valid and p ac cannot be calculated by a simple recipe described in the previous section. Fortunately, the following two relations enable one to express the pac•s of weighted graphs with loops in terms of graphs without loops and weighted edges. Let us consider a graph Gh with a loop of weight h attached to the vertex v . G1i may contain also other loops or weighted edges. Let G and G- v be obtained from Gh by deleting the loop at the vertex v and by deleting the vertex v, respectrvely. Proposition 5 P (Gh, x) = P (G, x)- hP (G-v, x) (16a) pac (Gh,X)=Pac (G,x)-hPac (G-v,x) * This graph is obtained by joining the central vertices of two chains with 2n -1 vertices. 94 I. GUTMAN ET AL. Proof. - Let S be the set of all Sachs graphs of the graph Gh. The Sachs graphs from S either contain the -loo·p h or not. Let the former comprise a subset Sh, while the latter a subset S_. Of course, Sh U S_ = S. On the other hand, ~ (- l)c (s) 2r (s) xN-n (s) = -h P (G-v, x) SE Sh ~ (- l)c (s) 2r (s) xN - n (s) = P (G, x) SES_ out of which eq. (16a) follows immediately. The proof of eq. (16b) is analogous. Let Gk be a graph with an edge e of the weight k. Gk may contain also other weighted edges and loops, except that the two vertices incident to e must not possess loops. Proposition 6 pac (Gk, x) = pac (G- e, x) - k 2 pac (G- (e), x) Proof - is analogous to that of Proposition 3. Propositions 5 and 6 present a relatively simple procedure for the cal-culation of pac's for graphs representing heteroconjugated systems. The graph of a quinoline-like molecule may serve as an illustration. Application of Proposition 5 gives P"( C9) " P"' (co) -hP0'( co) h while from Proposition 6, P" ( OG1) " P0 ' ( ()) ) - k2L8 ' p"' ( C0 ) -k' p"' ( v ) -k 2 Ls Hence, the problem is reduced to obtaining of pac's of simple graphs. Finally, p'"( cq) " Li Lg - L1 L3L, -k2 i 2L8 - L,L, I - hi 4J -L,L,I h Numerical work on the TRE's of heterocyclic compounds will be reported elsewhere26 • GRAPH THEORY AND MOLECULAR ORBITALS 95 REFERENCES 1. M. J. S. Dewar, Molecular Orbital Theory of Organic Chemistry, Mc Graw--Hill, New York 1969, Chapter 5. 2. J. Kruszewski and T. M. Kry go w ski, Tetrahedron L ett. (1970) 319. 3. B. A . Hess,_ Jr., and L. J. Schaad, J . Amer. Chem. Soc. 93 (1971) 305. 4. M. Mi 1 u n, Z. Sobotka, and N. Trina j st .i c, J. Org. Chem. 37 (1972) 139. 5. W. Eng 1 and aind K. Rued en berg, J. Amer. Chem. Soc. 95 (1973) 8769. 6. J.-I. Aihara, Bull. Chem. Soc. Japan 47 (1974) 3169; ibid. 48 (1975) 517. 7. C. F . W i 1 cox, Croat. Chem. Acta 47 (1975) 87. 8. P. Ge or g e, M. Tr a ch t m a n , C. W. B o ck, and A. M. B re t t , Theor. Chim. Acta 38 (1975) 121. 9. N. C. Baird, J. Chem. Educ. 48 (1971) 509. 10. M. J. S. Dewar a nd G. J . G 1 e i ch er, J. Amer. Chem. Soc. 87 (1965) 692. 11. B. A. Hess, Jr. and L. J. Sch aa d, Ibid. 93 (1971) 24i3. 12. B . A. Hess, Jr. and L. J. Scha a d, J. Org. Chem. 36 (1971) 3418. 13. L. J . Schaad and B. A . Hes s, Jr. , J . Amer. Chem. Soc. 94 (1972) 3068. 14. I. Gutman, M. Mi 1 u n , and N. T r ina j st i c, Croat. Chem. Acta 44 (1972) 207. 15. Part XIV: I. Gutman and N. Trina j st i c, Acta Chem. Acad. Sci. Hung., in press. 16. C. A. Co u 1 son, Proc. Cambridge Phil. Soc. 36 (1940) 201. 17. A . Gr a ova c, I. Gutman, and N. Trina j st i c, Chern. Phys. Lett. 35 (1975) 555. 18. Part XV : I. Gutm a n and N. Trina j st i c, J. Chem. Phys. 64 (1976) 0000. 19. Part I: A. Gr a ova c, I. Gutm a n, N. T ri n a j st i c, a nd T . Ziv k o vi c, Theor. Chim. Acta 26 (1972) 67. 20. I. Gutman and N. Trina j st i c, Topics Curr. Chem. 42 (1973) 49. 21. H . S a ch s, Publ. Math. (Debrecen) 11 (1964) 119. 22. H . Ho s o ya, BuU. Chem. Soc. Japan 44 (1971) 2332. 23. H . Hos c ya, Theor . Chim. Acta 25 (1972) 215. 24 .L. Lovasz and J. Pe 1 i k an, Period. Math. Hung. 3 (1973) 175. 25. E. He i 1 b r o n n er, H elv. Chim. Acta 36 (1952) 170. 26. I. Gut m a n , M. Mi 1 u n, and N. Tri n a j s t i c, in preparation. 27. R. B. Ma 11 ion, A. J. Schwenk, and N. T r ina j st i c, Croat. Chem. Acta 46 (1974) 171. 28. R . B . Mallion, N. Trin a j stic, and A. J. Sc h w enk, Z. Naturforsch. 29a (1974) 1481. SAZETAK Teorija grafova molekularne orbitale. XVIII. 0 topoloskoj r ezonancijskoj ener giji I. Gutman, M. Milun i N. Trinajstic Pokazano je kako se topoloska rezonancijska energija, TRE, m ofo prakticki iz-racunati. Definiran je aciklicki karakter isticni polinom, P "c, i aciklicki spektar gr afa, a takoder je diskutirana i njihova bliska veza s TRE. Prikazana je graficka metoda, koja omogucuje na vrlo jednostavan i elegantan nacin odredivanj e P"" i za jedno-stavne i za slozene molekularne grafove. INSTITUT »RUDER BOSKOVIC« I TVORNICA FARMACEUTSKIH I K E MIJSKIH PROIZVODA »PLIVA«, 41000 ZAGREB Prispj elo 2. stude n og 1975. 

Graph Theory and Molecular Orbitals. XVIII. On Topological Resonance Energy

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