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IJCA 32A(10) 833-836.Pdf Indian Journal of Chemistry Vol. 32A, October 1993, pp. 833-836 Approximating the total .7l-electron energy of benzenoid hydrocarbons: 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<ji: embraces p = (v)2 terms. Bear- creasin order. The total Jr-electron ener (when expressG in t dusualin themanner2.16•units of theThenHMOG hasreson~verticesnce in• ing this in mind, we can rewrite formulae (3) and tegral f3 satisfies the relation (5)as, a/p = «(xjxY) ... (6) 2 Xl, if n is even and E= I' n-')!2 E=2[m+2p(xjx)]I/2 ... (7) where (.) denotes the arithmetic mean of the re• 2 r2I x, + X(n+1)/2 if n is odd ! 1=1 spective variable. 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 <j ,1.,(5) number of Kekule structures, and combining Eqs 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, benzene 0= (n - 2)/m .... (17) (K = 2), naphthalene (K = 3), anthracene (K = 4), Recall that a connected graph on n vertices has at phenanthrene (K= 5), naphthacene (K= 5), pyrene least n - 1 edges. (K = 6), pentacene (K = 6), benz[ a]anthracene From (7), (16) & (17), the inequality (K = 7), hexacene (K = 7), chrysene (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.
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