A symmetric two-delta-function model spectral density is used
to estimate several topological indices of alternant hydrocarbons,
namely: the total n-electron energy (E.), the modified topological
index (Z), the HOlVIO-LUMO separation (XHL) and the spectral
radius of adjacency matrix (R). It is found, that the invariants defined by integration (like E. and Z) are reproduced much better than the invariants defined as the Iimiting values of the spectral distribution (like XHL and R). The reason for the well known linear dependence between Er. and lnZ, is discussed

A. Graovac

J. Cioslowski

English

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CROATICA CH,EMICA ACTA CCACAA 61 (4) 797-808 (1988)CCA-1827YU ISSN 0011-1643UDC 541.17Original Scientific PaperAn Approximate Spectral Density for the Estimation of someTopological Indices of A1ternant SystemsA. Graovac' and J. Cioslowski2MARS, TAMUG, Galveston, Texas 77553; USAReceived September 7, 1987A symmetric two-delta-function model spectral density is usedto estimate several topological indices of alternant hydrocarbons,namely: the total n-electron energy (E.), the modified topologicalindex (Z), the HOlVIO-LUMO separation (XHL) and the spectralradius of adjacency matrix (R). It is found, that the invariantsdefined by integration (like E. and Z) are reproduced much betterthan the invariants defined as the Iimiting values of the spectraldistribution (like XHL and R). The reason for the well known lineardependence between Er. and lnZ, is discussed.INTRODUCTIONThe description of conjugated hydrocarbons through the concepts ofchemical topology and graph theory is well documented to be an effectiveway to account for their physicochemical properties.' In particular, the totalrc-electron energy (E,,) was a subject of several theoretical approaches.A great deal of attention has been focused recently on the spectraldensity formalism for the calculation of topological indices of alternantsystems.v" Let the eigenvalues of the adjacency matrix describing the systemunder study that has N carbon atoms, M carbon-carbon bonds and K Kekulestructures, be ±x), ±X2,"" ±XN/2. The spectral density function is definedas:"N/2I' (x) = ~ [Il (x - x.) + Il (x + Xi)]i= 1(1)The spectral density comprises all the information about eigenvalues and.therefore, the knowledge of its form is sufficient to calculate several topo-logical invariants. The exact function is usually approximated by somemodel density D (x). Both discrete" and continuous" forms of D (x) wereproposed in the chemical literature. Recently, some very general propertiesof D (x) have been deduced.2,3,4During the derivation of D (x) the normalized invariants appear in anatural way. They are:"1 Permanent address: »Rudjer Bošković« Institute, YU-41001 Zagreb, POB 1016,Croatia, Yugoslavia2 Present address: Department of Chemistry, The University of North Carolinaat Chapel Hill, Chapel Hill, North Carolina 27514, USA.798 A. GRAOVAC AND J. CIOSLOWSKI1. The McClelland quotient (MCQ):e = E, (2MNftj,; (la)where E" is the sum of the moduli of eigenvalues.2. The normalized structure count (NSC):k = K2jN (2M/NflJ,; (lb)In general case the number of Kekule structures should be replaced by thealgebraic structure count.3. The normalized HOMO-LUMO separation (NHLS):h = 1/2 XHL (2M/Nftj,; (lc)where XHL is twice the eigenvalue with the smallest modulus.4. The normalized spectral radius (NSR):r = R (2M/Nr'j,; (ld)where R is equal to the eigenvalue with the largest modulus.The simplest form of D (x) is a normalized sum of two Dirac's functions:D (x) = N/2 [1)(x-a) + 1)(x + all. (2)Such a choice leads to e = k = h = r = 1 and, therefore, is too crude todescribe more subtle effects of the molecular topology on E" and XHL. Inthe present paper we investigate a slightly more sophisticated form of D (x)(Figure 1):D (x) = N/4 [<5 (x - a) + <5(x + a) + <5(x - b) + o (x + b)],which is essentially a symmetrized sum of two Dirac's delta functions.D (x) = [<5 (x + b) + <5(x + a) + <5(x - a) + <5(x - b)] . N/4-b -o O +0 +b xFigure 1. A symmetric two-delta function spectral density.THE ORYThe parameters a and b can be computed easily from the conditions (5)and (6),3 while the normalization factor of N/4 guarantees equation (4) tobe fulfilled:+00S D (x) dx = N, (4)-00TOPOLOGICAL INDICES 799+00S D (x) X2 dx = 2M, (5)+00S D (x) in Ix! dx = 2 in K. (6)-00From these constrains we calculate a and b as:a = d'/, [1 - (1 - k4)'/'l'/',b = d';, [1 + (1 - k4)1/,l'/',where d is the mean vertex degree of the molecular graph:d = 2M/N(7)(8)(9)The quantities a and b can be regarded as estimates for the HOMO-LUlVIOseparation and the spectral radius. Bearing in mind definitions (lc) and (ld)we can write:(10)and(11)The total rc-electron energy depends also on D (x) in a straightforward way:+00S D (x) I x I dx= E". (12)Within the approximation (3) this gives rise to:z;= (2MN)'/' [(1 + k2)/21'/' (13)ar:e= [(1 + k2)/21't.. (14)The approximation (13) was first considered by Gutman et aU Eventually,we can compute the modified topological index Z from the relation:"+00S D (x) in (1 + x2) dx= 2 in Z, (15)-00which results in the approximate relation:in Z= N/41n (1 + 2d + d2 k4). (16)DISCUSSION AND NUMERICAL CALCULATIONS""-TiFirst we discuss the equations (10), (11) and (14) that express anapproximate dependence of h, r and e on k. The very fact that h, r ande depend only on k is not accidental but it is acorollary of the theorerrr'which states that any approximate formulae expressing E", XHL and R asthe functions of N, M and K must be reducible to simple relations betweenthe normalized topological invariants and NSC.The modified topological index Z is a different case. Inspection of eq.(16) reveals that Z is not a simple function of k. On the first glance it is800 A. GRAOVAC AND J. CIOSLOWSKIin apparent contradiction with the well known faet that E~ and In Z arerelated through a approximate line ar regression.v P This paradox can beresolved in the following way:Let us rewrite eq. (16) as:z = (2 MNr'f, In Z= (16dr'!' In (1+ 2 d2 k"), (17)Ii the assumption on the approximate linear relation between E" and In Zis valid, one expects that the ratio:zle = (8 dr'!, (1+ k2r'!2 In (1+ 2 d + d2 k") (13)varies only very slightly with both k and d. Since d takes the valuesbetween 2 and 3 and k between 0.7 and 0.9,3 we carried out simple testcomputations of the ratio for these limiting values. The results, presentedin Table I, show that the ratio is indeed almost constant, at least for d andk lying with in the above limits.TABLE IThe Ratio zi e for Various VaLues of d and k (eq. 18)dk2.0 2.5 3.00.7 0.3656 0.3691 0.37040.8 0.3695 0.3749 0.37760.9 0.3774 0.3844 0.3831We, thus, conclude that the linear relation between Er; and In Z originatesfrom' an accidental numerical independence of the ratio on d and k ratherthan from more general reasons.The accuracy of approximate formulae (7), (8), (13) and (16) has beentested against the standard set of benzenoid hydrocarbons (1030 moleculesj.P."The results of the testing are shown in Table Ir.TABLE IILinear Regression Between Approximate and Exact TopoLogicaL Invariants'I'opological invariant Er. Z EHL RLinear regression :Slope: 0.999 0.998 2.865 0.987Intercept: 0.239 0.217 -1.793 0.571Errors:RMS 0.3°/0 0.60f0 100f0 1.10f0Standard deviation: 0.50f0 0.9% 16% 1.4~/oMaximal: 2.5% 4.3% 131% 7.9%Correlation coefficient: 0.9990 0.9962 0.9151 0.4300TOPOLOGICAL INDICES 801The exact total 7t-electron energy is reproduced by our simple modelrather accurately (Figure 2). We, therefore, conc1ude that this quantity israther insensitive on the shape of spectral density, provided the approxi-mation to density fulfills the constrains (4)-(6). The same statement is validfor the modified topological index. It should be pointed out that in all thecalculations the benzene molecule was exc1uded, because of the degeneracyof HOMO and LUMO orbitals. The degeneracy prec1udes the use of a simpletwo-delta approximation to the density function.xo,o,ot=W40302020 30Ere (exo ct )40Figure 2. The approximate versus exact values of the total Jt-electron energy forthe standard set of benzenoid hydrocarbons. The results are given in ~ units of theHMO theory.The situation with the HOMO energy, €HL = 1/2 XI!L, and the spectralradius, R, is quite different. There is a definite correlation between thecalculated and exact €HL, but the dependence has rather a qualitativecharacter (Figure 3). The trend in HOMO energies is well accounted for,but the slope of regression is far from unity. Eventually, there is nomeaningful correlation between the calculated and exact R.The simple model for spectral density function presented in our paperis intented to show how some trends in topological indices can be analyzedanalytically with the knowledge of very limited information about moleculargraph rather than to produce very accurate formulae for these indices. Itseems to be c1ear that the two-delta approximation is well suited for suchan analysis.802 A. GRAOVAC AND J. CIOSLOWSKI0.80- 0.75xQ.Q.o...JIW<l 0.700.65.'I .:' ~ v.c."'" ..'ff' ~t:• ·CI,It\" "a •1-' ,. II~O ._\ _._.u..:.:.~: ..e_~. ~.-11 •'H ,.•.;r-',. V •..... ,. ....-."'.:~'. '- ..::;": ..I., .-&.::. . .•....... . .•• o .......0.6Figure 3. The approximate versus exact values (in ~ units) of the HOMO-LUMOseparation for the same set of benzenoid hydrocarbons as in Figure 2.0.1 0.2 0.3 0.4 0.5AcknowLedgement. - One of the authors (A. G.) thanks the Welch Foundation,Houston, Texas, for a grant.REFERENCES1. See for example: 1. G u t man and O. E. P o 1a n sky, MathematicaL Con-cepts in Organic Chemistry, Springer- Verlag, Berlin 1986 and reference s citedtherein.2. J. C i o s 1o W s k i, MATCH 20 (1986) 95.3. J. C i o s 1o W s k i, Int. J. Quant. Chem. 31 (1987) 581.4. J. C i o s 1o W s k i and O. E. P o 1a n sky, TheoT. Chem. Acta (Bert.), 74 (1988)55.5. 1. G u t man, MATCH 14 (1983) 71.6. 1. G u t man, and M. R a š k o v i ć, Z. Naturforsch. 40a (1985) 1059.7. 1. G u tm a n, L j. Ne del j ko v i Ć, and A. V. Te o dor o v i Ć, BuR Soc.Chim. Beograd 48 (1983) 495.8. H. H o s o y a, K. H o s o i, and 1. Gu t man, Theor. Chim. Acta 38 (1975) 3'l.9. J. Ai h ara, J. Ort), Chem. 41 (1976) 2488.10. 1. G u t man, A. V. Te o dor o v i Ć, and Z. B uga r č i Ć, Bu!I. Soc. Chim.Beograd 49 (1984) 521.11. J. C i o s 1o W s k i, MATCH 20 (1986) 103.12. J. C i o s 1o W s k i, Z. Naturforsch. 40a (1985) 1169.13. J. C i o s 1o W s k i, Theor. Chim. Acta (BerL) 68 (1985) 315.14. J. C i o s 1o W s k i, J. Comp. Chem. 8 (1987) 906.TOPOLOGICAL INDICES 803SAŽETAKProcjena nekih topoloških indeksa u alernantnim sustavima pomoću jednepribližne spektraIne gustoćeA. Graovac i J. CioslowskiSimetrična spektralna gustoća modelirana dvjema delta-funkcijama upo-trijebljena je za procjenu slijedećih topoloških indeksa u alternantnim ugljiko-vodicima: ukupne .n-elektronske energije (En), modificiranog topološkog indeksa (Z),HOMO-LUMO separacije (XHL) i spektralnog polumjera (R) matrice susjedstva.Nađeno je da procjena bolje reproducira indekse vezane uz integraciju spektralnegustoće (En, Z) nego one (XHL, R) vezane uz lokaciju delta-funkcija. Prodi skuti ranisu i razlozi za valjanost dobro poznate linearne ovisnosti En O In Z.

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