Indian Journal of Chemistry Vol. 32A, October 1993, pp. 833-836

Approximating the total .7l-electron energy of benzenoid : Some new estimates of (n,m)-type \I Ivan Gutman* Institute of Chemistry, Academia Sinica, Taipei 11529, Taiwan, ROC and Lemi Turker Department of Chemistry, Middle East Technical University, Ankara, Turkey Received 15 February 1993; accepted 3 May 1993

Three novel lower bounds for the total .7l-electronenergy of benzenoid hydrocarbons have been obtained. These bounds are of (n,m)-type, i.e., they are functions of only the number of carbon at• oms (n) and the number of carbon-carbon bonds (m). Two of them are much better than the previ• ously reported estimates of the same kind. The results obtained shed light on the dependence of the thermodynamic stability of benzenoid hydrocarbons on molecular structure. The extension of the re• sults obtained to non-benzenoid altemant hydrocarbons is straightforward, but requires some cau• tion.

The total Jl-electron energy (E), as calculated carbon-carbon bonds. This finding had two signi• within the framework of the tight-binding Huckel ficant consequences. First, it highlighted the im• molecular orbital model, is a quantum-chemical portance of the structural parameters n and m. characteristic of conjugated hydrocarbons (and Second, it shifted the focus of research from find• benzenoid systems in particular) that provides ing approximate formulae for E towards finding useful information about their thermodynamic be• mathematically exact upper and lower estimates haviour. The fundamental problem of how E is for E. related to the (experimentally measurable) heats Since McClelland's seminal paper4, a large of formation and atomization was resolved by number of results in the same field were ob• Schaad and Hess I, who showed that the total en• tained5-1z (see reference 3 for review and further ergy of the a-electrons of the. carbon-atom frame• references). In particular, several (n,m)-type lower work also depends on E, and that, in the absence and upper bounds for the total Jl-electron energy of steric strain, this dependence is linear. Conse• of benzenoid hydrocarbons were designed3. A quently, there exists a linear relation between E general conclusion of these studies is that the par• and experimental enthalpies, especially in the case ameters nand m are responsible for about 99.5% of benzenoid hydrocarbons (see references 2 and of E, all other structural details having an effect 3 for further details). of less than 0.5%. [The most important of these Early theoretical studies on E were aimed at structural details were recently identified13-15 to be finding methods for its easy calculation. In view of the Kekule structure count and the number of bay the rapid development of computational chemis• regions.] try and the ubiquity of personal computers, this In this paper we report three novel lower task can nowadays be considered as largely obso• bounds for E of (n,m)-type, two of which are lete. A related problem, however, the establishing found to be significantly better than the previous• of how E of a molecule depends on its structure, ly known bounds of the same kind. In order to continues to attract attention of researchers and is formulate our results we need to introduce a few far from. being completely solved. A major impe• notions from the graph-spectral theory of conju• tus in this direction was the discovery by McClel• gated moleculesZ•16• land4 of the inequality E ~ j2mn, where n is the number of carbon atoms and m the number of Some definitions from chemical graph theory and ail identity for E tPerrnanent address: Faculty of Science, University of Kra• Let the conjugated molecule being considered gujevac, P.O. Box 60, YU-3400 Kragujevac, Yugoslavia be represented by the molecular (Huckel) graph l

.,~ H34 INDIAN J CHEM, SEe. A, OCTOBER 1<)<)3

and m dges. The eigenvalues2 of G wi! be de• noted x, , x2, ..., xn and are labelled in non-in• The summation i

Thro out this paper, it is assumed t at n is an even number and that Xn/2 > O. This i tanta• mount t restricting the consideration to benze• noid hy rocarbons having a closed-shell round• state Jr- lectron configuration 1(,. [The cases of odd nand n/2 = 0 (which are chemically mu h less relevant can be treated in a fully analogo s man• ... (8) ner, bea 'ng in mind that for the molecular graphs

of alter ant hydrocarbons the eigenvalue x(n+1 )/2 is neces arily equal to zero, and that for benze• and that the summation I goes over noid hy rocarbons, the condition xn/2 = () mplies i.j.k,1 the lack f existence of Kekule structures.] Hence q = v( v-I) (v2 - v- 2)/8 summands. Conse- e that E can be written in the form, quently,

... (1) (xixjxkx,)=(lIq) I x,XjXkX,. ., . (9) i.j.k,1

The n mber of edges of the molecular g ph of Now, because the geometric mean of non-nega• tive numbers cannot exceed their arithmetic mean, we get eigenvalan altern tants hydrocarbonin the followingis related towell~ownth:r;raPh manner2.where V~n!2'h: .,. (10) ... (2) Every graph eigenvalue occurs r=(v-l)(v2- v-2)/2 polynomi~FIf a4 is t~e fourthof the molecularcoefficientgraphof theG, characteristicthen 7 times in the product n XiXjXkX]. Hence, the i,j,k,l ... (3) right-hand side of Eq. (10) is equal to,

. ., (11) [Vj~ Xi]f/q = [Vi~ Xi]41V = Idet A/2/v minedFor benzby noidthe parametershydrocarbons,nand a4 m,is andfullysatisfies~eter• the identi 17, where det A= n Xiis the determinant of the i~l

a4=(m2-9m+6n)/2.Combi ing (1) and (2), one of the presett1 .. au•(4) thors ded ced7 the following identity for E: adjacency matrix A of the molecular graph. Bear• ing in mind! the Dewar-Longuet-Higgins rela• tion I x" Y det A=(- 1)~ 2, which holds for benze• noid hydrocarbons and in which K denotes the E=2[m 2IXlxJ]'2I

Itli 'I f 1·11 GUTMAN el al.: .7l-ELECTRON ENERGY OF BENZENOID HYDROCARBONS 835

(4), (5) & (8 )-(11),we arrive at the inequality, We want to fir.J an expression for 0 in terms of E ~[4m +[32(m2-9m +6n)+n(n - 2) the structural invariants n and m. The simplest x (n2- 2n - 8)K 8/n]1 12]1 12 .••• (12) such expression, which is smaller than unity for all molecular graphs (or more precisely, for all Now, there exist only twelve Kekulean benzenoid connected graphs) is, hydrocarbons for which the Kekule structure count K is less than nine, namely, 0= (n - 2)/m .... (17) (K = 2), (K = 3), (K = 4), Recall that a connected graph on n vertices has at (K= 5), naphthacene (K= 5), least n - 1 edges. (K = 6), (K = 6), benz[ a]anthracene From (7), (16) & (17), the inequality (K = 7), (K = 7), (K = 8), ben• zo[c]phenanthrene (K = 8) and heptacene (K = 8). ... (18) Keeping these exceptions in mind, we can trans• E~2 [m+ n: 2 [12n(n-2)a4 ] 1/2 ] 1/2 form relation (12) into, E ~[4m +[32(m2- 9m +6n)+n(n - 2) follows, that appears to apply to all alternant hy• x (n2 - 2n - 8)98/n]1!2]1!2 ••• (13) drocarbons. Combining it with (4), we obtain our second lower bound for E of the (n,m)-type: which is our first lower bound for E of the (n,m)• type. Numerical testing reveals that of the above listed 12 benzenoid hydrocarbons for which E~2 m+2 n m - n(n-2)(m2-9m+6n) [ 1 - ') [ ] 1/2 ] 112 K < 9, the relation (13) is not satisfied for only four (benzene, naphthalene, anthracene & phen• ... (19) anthrene). Hence, the relation (13) holds for all Numerical testing of (17) shows that for Kekulean Kekulean benzenoid systems with more than benzenoid hydrocarbons (but not for all alternant three hexagons. hydrocarbons) the inequality (19) can be further The second and third lower bounds sharpened by using 0' instead of 0, where Comparing the identities (5) and (8), we imme• o'=(n -1)/m .... (20) diately see that a lower bound for E is obtained if From (4), (7), (16) & (20) we gain an additional on the right-hand side of relation (8) we negle<;t lower bound for E of the (n,m)-type: some positive terms. If the neglected term is the

12 E~2 m+2~ n(n-2)(m2- 9m+6n) sum I,then we arrive at a previously known [In - 1 [ ] )/2 ] )12 i,j,k.l ... (21) ((n,m)-type inequality: E ~ 2[m + 2 [a4]I/2p/2= 2[m + [2m2 -18m The quality of the estimates (13), (14), (19) & + 12np/2]1/2.... (14) (21), as well as of the heuristic formulae (17) & (20) is examined in the following section. As shown in the subsequent section, the bound (14) is not very sharp. A somewhat better estim• Numerical work and discussion ate of the same kind is obtained by starting from The numerical work reported in this section the relations (6) and (7). The idea is to substitute was done on the standard data base consisting of (Xix) in relation (7) with a term that is smaller, 106 Kekulean benzenoid hydrocarbons taken but not too much smaller. Such a term may be of from the book of Zahradnik and Pancir20; the the form (6), Now, the arithmetic mean of some same set was used in our previous studies on total numbers never exceeds the square root of the J'C-electronenergy3. arithmetic mean of the square of these numbers. The functional dependence of the multiplier 0 In our case, on the parameters n and m, as given by Eq. (17), (XiXj>~[«(XiX/>]J/2=Ja4/p .... (15) was chosen because of its simplicity. We, how• In order to "invert" the inequality sign in relation ever, tested' the expression 0=x(n-2)/m and (15) we have to multiply its right-hand side by a found that the inequality (16) is satisfied only if suitably chosen quantity 0: the value of x does not exceed 1.03. Hence, the initial choice (x = 1) was an acceptable one. We al• (XiXj>~ 0 [«(XiXj/>t2 = 0 Ja4 /p .... (16) so tested the expression 0= [(n - 2)1m]Y and found Obviously, 0 must be < 1. that y must not be less than 0.85. This is again 836 INDIAN J ('HEM, SEe. A, OCTOBER 199]

(n,m)-type, but will depend on additional structu• relative d erences between the total ,n-electron e ergies of ral paramt~ters7.17. The condition Idet AI ~ 92, benzenoiTable I1veragehydrocarbonsrelativeanddifferencestheir (n,m)-typeand maximal10bservedlowe bounds; which was used in the derivation of (13), is violat• data base taken from ref, 20 ed by only a very small number of Kekulean Formula 53.1110.1Maximal13.64Average15.3762.24.]19.218.43,982.14 benzenoids .. On the other hand, the number of difference (%) difference (% (19)(22)((13)*14) (21 ) non-benzenoid species that violate this condition is unlimitedly large. It is, therefore, safer to as• sume Idet AI ~ 1, which when inserted into (12) yields the following variant of the estimate (13):

E ~ [4m + [64a4 + n(n - 2)(n2 - 2n - 8)]I/2r/2: .. (23) Relation (23) holds for alternant non-benzenoid *benzene, I naphthalene, anthracene & phenanthrene disre• garded hydrocarbons that possess a closed-shell ground• - state n-electron configuration. Note that for these . se to the initial guess y = 1. Fi ally, by molecules the identity (4) need not be applicable. examlill g the expression 0= (n - 2z)/m w found Finally, it seems that it is legitimate to apply that the inequality (16) is not violated un il z be• formula (17) to any conjugated . Its comes I ss than 0.40. This critical value or z is improvement, Eq. (20), needs to be separately signific tly smaller than the initially ssumed tested for each particular class of non-benzenoid one, na ely z = 1. Based on this observa ion we systems. It is, for example, certainly not applicable introdu ed the multiplier 0', Eq. (20); the expres• for acyclic conjugated molecules, because for sion gi en by formula (20) corresponds to the them m = n - 1 and consequently, 0' = o. choice x = y = 1, z = 0,5. Acknowledgement and (21 , which are reported in this paper for the One of the authors (I.G.) thanks the National firstIn timad irion, we alsoto theexaminedthree lowerthe previouslbounds (j3),known(19) Science Council of the Republic of China, for fi• (n,m)-ty e lower bounds (14 )12 and (22)11: nancial support.

7) 1/'"y2mn~ .... (22) References Its obtained are summarized in Tab e 1. 1 Schaad L J & Hess B A, J Am chem Soc, 94 (1972) The ta presented in Table 1 clearly s ow that 3068, the low r bounds (19) and (21) provide a signifi• 2 Gutman I & Polansky 0 E, Mathematical concepts in or- ganic chemistry (Springer-Verlag, Berlin), 1986, cant i provement relative to the pr viously 3 Gutman I, Topicscurr Chem, 162 (1992) 29. known esuIts in this area. Formula (31) as ob• 4 McClelland B J, ] chem Phys, 54 (1971) 640, tained a trial-and-error adjustment of t e form 5 Gutman I, Chern Phys Leu, 24 (1974) 283, of the nction 0= o(n,m). It is hoped t t even 6 Bochvar D A & Stankevich I V, Zh strukt Khim, 21 better e pression for this functional dep ndence (1980)61. 7 Tiirker L, Math Chem, 16 (1984) 83. will be ound in the future. This task is p obably 8 Cioslowski J, Z NatUlforsch, 40a (1985) 1167, closely elated to the problem of conc iving a 9 Gutman I, Tiirker L & Dias J R, Math Chem, 19 (1986) mathem tically satisfactory proof of the in qualit• 147. ies(19) d(21). 10 Cioslowski J, Intern] Quantum Chern, 34 (1988) 217, The r suIts of this paper are formulated for Ke• 11 Gutman 1,.1chern Soc Faraday Trans, 86 (1990) 3373. 12 Tiirker L, Math Chern, 28 (1992) 261; Gutman I, Math kulean enzenoid hydrocarbons. Their e tension Chem, 29 (1993) 61. to odd enzenoids and non-benzenoid m lecules 13 Hall G G, Intern] Quantum Chem, 39 (1991) 605. is straig tforward, but requires a certain d gree of 14 Gutman I, Hall G G, Markovic S, StankoviC Z & Radi• caution. First of all, if n is odd, then v sh uld be vojevic V, Polyc arom Comp, 2 (1991 ) 275, identifie with (n - 1)12. In that case, th main 15 Gutman I & Hall G G, Intern J Quantum Chen!, 41 (starting equations of this paper remain v lid if n (1992)667, 16 Graovac A. Gutman I & TrinajstiC N, Topological arr is eve here exchanged by n - 1. proach to the chemistry of conjugated molecules (Spring• The y relation which is fully inappli able to er-Verlag,Berlin),1977, non-be enoids is Eq. (4). Thus, the term 4 must 17 Gutman I, Croat chern Acta, 46 (1974) 209, remain i the respective expressions, as fo exam• 18 Dewar M J S & Longuet-Higgins H C, Proc roy Soc Lon• ple in (1 ). As a consequence, in the case f non• don, A214 (1952) 482. 19 Gutman I, Topics curr Chem, 162 (1992) L benzeno ds, the lower bounds obtained by the 20 Zahradnik R & Pancir J, HMO energy characteristics (Ple• presentl described procedure will not be of num Press, New York), 1970. "