Some further classes of isomers exibiting the Topological Effect on Molecular Orbitals (TEMO) are considered. They include two classes of hetero-substituted polyenes and three classes of substituted annulenes. The sixth class contains bridged annulenes where TEMO is exhibited only for the related matching spectra

Ante Graovac

Jahi Hoxha

Oskar E. Polansky

English

Hrčak - Portal of scientific journals of Croatia

CROATICA CHEMICA ACTA CCACAA 59 (3) 591-598 (1986)CCA-1671YU ISSN 0011-1643UDe 541Original Scientific PaperOn Some Further Classes of Isomers Which ExhibitTopological Effect on Molecular Orbitals*Jahi HoxhaDepartment of Physics, Faculty of Science, University of Kosovo, Trg Maršala Titabb, 38000 Priština, Kosovo, Serbia, Yugoslavia,Ante GraovacRugjer Bošković Instite, P.O.B. 1016, 41001 Zagreb, Croatia, YugoslaviaandOskar E. PolanskyMax-Planck-Institut fur Strahlenchemie, Stiftstr. 34-46, D-4330 MULheim/Ruhr,F. R. GermanyReceived October 21, 1985Some further classes of isomers exibiting the TopologicalEffect on Molecular Orbitals (TEMO) are considered. They includetwo classes of hetero-substituted polyenes and three classes ofsubstituted annulenes. The sixth class contains bridged annuleneswhere TEMO is exhibited only for the related matching spectra.INTRODUCTIONThe regularity of the molecular orbital (MO) energies arrsmg form thedifferent topologies of isomers is called the »Topological Effect on MolecularOrbitals (TEMO)«1. The effect has been confirmed by experimental findings",as well as by quantum-chemical computational procedures of various levels ofsophisticationš-!The already existing cIasses of isomers which exhibit TEMO are extendedin the present paper by a few further cIasses. They incIude two cIasses ofhetero-substituted polyenes and three cIasses of substituted annulenes. Thesixth cIass contains bridged ann ul enes where TEMO is exhibited only forthe reI at ed matching spectra.The mathematics of TEMO is relatively simple. Different topologies ofthe isomers MA and MB under consideration are appropriately described bythe graphs A and B associated with the isomers", The corresponding charac-teristic polynomials ([> (A; x) and ([> (B; x) are then compared. There are vari-ous cIasses of isomerst= for which the difference function, il (B, A; x), satisfies:t.. (B, A; x) == ip (B; x) - ip (A; x) 2: O (1)* Presented at The International Symposium on Applications of MathematicalConcepts in Chemistry, Dubrovnik, September 2-5, 1985.592 J. HOXHA ET AL.The above inequality implies'A?X2k_l (A) 2:: x2k-l (B) 2:: x2k(B) 2:: x2k (A) 2:: .•. (2)where Xj (A) and Xj (B), j = 1, 2, ... , N, denote the roots of (JJ (A; x) and(JJ (B; x), respectively, i. e. the MO energies OI the isomers MA and MB as cal-culated within the simple MO theory. N stands for the number of conjugatedcenters in the considered isomer and equals the number of vertices in theassociated graph.The interlacing rule (2) is called TEMa! and it enables predictions onthe first absorption maxima, ionization potentials, etc., of the isomers MA andMB. In the case where the sign of ~ (B, A; x) generally changes along theenergy parameter x-asis, TEMa with inversion- holds.HETERO-SUBSTITUTED POL YENESA linear polyene, CH2(CH)N_2CH2,may be rep resen ted by the path graph,PN, with N vertices. Let us introduce two heteroatoms into the polyene atthe positions i and j. Let hi and h2 stand for the values of the Coulombintegral of the first and the second heteroatom, respectively. The correspond-ing (weighted) graph is denoted by (PN;i, j; hit h2)'Class 1. - Let N be an odd number, N = 2n-1. Let consider the hetero--substituted polyene MA with the first heteroatom at the position n in themiddle of the polyene and with the second heteroatom at the position 1,1> n. The related graph A = (P2n-1;n, 1;hit h2) is shown in Figure 1. Hereand in the further figures black a circle represents a heteroatom. Let usfurther consider the isomers of MA in which the first to heteroatom is movedout of the middle position to the new position 1 + P while the second hetero-atom remains at its position 1. These isomers are represented by Bp == (P2n-1;n + p, 1;hit h2), p = 1, 2, ... ,1- n-I, and a representative is de-picted in Figure 1.(}-D---O- _1 2 3hl---0---4.)--0- _n-l n n-]h2-0- --0l 2n-,cr-o--o- _2 3h·1--0----0---0--- -0- -0- __~n-l n n+' n+p 2n-lTEMO IN SOME ISOMERS 593The third factor on the right-hand side of eq. (3) is a nonnegative quantity.However, the parity of the second factor changes along x-axis and TEMOwith inversions holds. The inversion points are easily calculated after thevalues of 1, p and h2 are given and they don't depend on the length, 2n - 1,of the considered hetero-substituted polyenes.If the heteroatom h2 is moved to the left side, p < 0, of the middleposition, an equation analogous to eq. (3) can be derived by using the fol-lowing well-known identity:(4)The same comment applies to the rest of this paper as well.Class 2. - Let us now consider the situation where both heteroatomsare moved simultaneously by the same amount, p.Let N be an even number, N = 2n. Let us consider the hetero-substitutedisomer MA whose graph is A = (P2n; n, 1;ht, h2). Let us further consider theisomers M s, of MA which are represented by the graphs Bp= (P2n; n + p,1+ p; ht, h2). The difference function is rather complicated. In the special case1= n + 1, i. e. where two heteroatoms are neighbours in MA and MBp , andfor both heteroatoms being of the same kind, hi = h2 = h, the differencefunction is simplified to the following form:(5)p = 1, 2, ... , n - 1The corresponding graphs are depicted in Figure 2.0---0---0-- _1 2 3h h• • --02n0---0- --0-- _1 2 3h h-0---0-___ _ • -0n n+l n s p n-p-t 2nFigure 2.Eq. (5) implies that TEMO with only one inversion point Xinversion = h,holds.In the case of electron attracting heteroatoms, h > 0, the inversion pointlies in the region of the bonding MO's. Up to the inversion point the regu-larity (2) holds where B stands for any of Bp's.In the case of heteroatoms with h< 0, the inversion point lies in theregion of the anti-bonding MO's. The regularity (2) holds in the whole regionof bonding MO's but the symbols A and B have to interchange their placesbecause !:J. (A, Bp;x) ::; ° for x ;?; o.594 ;T. HOXHA ET AL.SUBSTITUTED ANNULENESAn annulene, CNHN, may be represented by the cyc1e graph, CN, withN vertices.Further, we consider three c1asses of substituted annulenes.Class 3. - Let us represent the graph attached to the annulene CNHNby graph F. The linking site of the attached graph is represented by thevertex f. The graph obtained by deletion of f out from F is denoted by F- f.Let us consider a substituted annulene where two graphs FI and F2 areattached at the positions i and j, respectively. The corresponding graph isdenoted by (CN;i, j; r; F2)'Let N be an even number, N = 2n. Let us further take: FI = F2 = F. Letus consider the substituted annulene MB which is represented by graphB = (C2n; n, 2n; F, F) of Figure 3. If the graphs are attached at arbitrarypositions, a series of isomers MA, is obtained which are represented by graphsAl = (C2n; 1,2n; F, F), l = 1, 2, ... , n - 1. A representative of the series isdepicted in Figure 3.2n -1//(I/I,,,,,II/~n.l n n-lB=(C2n;n,2njF, F)Figure 3.The difference function is non-negative for the whole series:ep (B; x) - ep (Al; x) = 3 [ep (F - f; x»)2 • [ep (Pn-I; X)]2 > O (6)I = 1,2, ... ,n-lConsequently, TEMa with out inversions (2) applies if MO energies of anyof isomers MA, are compared with MO energies of the starting isom er MB.Let us note that B and Ar's together represent n isomers. By using theterminology of TEMOl Ar, A3, ••• , represent S-isomers of Model 23 as theypossess the planes of symmetry kr, k2' . .. . Similarly, A2, A4, ••• , representS-isomers of Model 33 as they possess the planes of symmetry O"r, 0"2, ••• , which•TEMO IN SOME ISOMERS 595pass through a pair of vertices. Altogether, there are (n - 1) S-isomers, [nI2][n-I]of them are of 1:-type and -2- of c-type. Brepresents T-isomer1 withregard to any of (n -1) S-isomers. The situation is c1arified in Figure 4.L2 )IIIIn~/n+J+l nAlA2BFigure 4.Class 4. - Let us consider substituted annulenes whose graphs are-shown in Figure 5. Let further N be an even number, N = 2n.2n-1-,,/IIII~.n+1 n n-1Figure 5.20Jt,,Ih\IHI//~.n+1 n n-1II~1t,t~The difference function for the isomer represented by B and for S isomerrepresented by some of Al'S is given by:1== 1,2,. o o, n-Iij) (B; x) - ij) (AI; x) ==h o ij) (F - f; x) o [ij) (pn-I; x)]2 (7)'596 J. HOXHA ET AL.-Consequently, TEMO with inversions holds where the inversion points arethe roots of ep (F - i, x) which is easy to calculate after the attached graphrepresented by F is specified.Class 5. - The isomers of this class can be understood as a special caseof Class 3 isomers. They contain two, generally different, heteroatoms withthe values hI and h2 associated with them. N is as before an even number,N = 2n. The isomers are rep resen ted by graphs which are depicted in Fi-:gure 6.2n2n-l~/ -' .•..•.:' \"2n2n-l~/' -'''',I \1! 91"~n+1 n n-1JI"I/.~n+1 n n-1Figure 6.'The difference function is easy to ca1culate and is given byp (B; x) - p (Al; x) = hI hz [P (Pn-'; x)]2 (8)I = 1, 2, ... ,n-IAs a result, TEMO as is given by (2) holds if both hI and hz are of the sameparity. If hi and hz are of opposite parity, TEMO again holds but in (2) onehas to interchange the symbols B and A. Obviously, A stands for any of AI's.MATCHING SPECTRA OF BRIDGED ANNULENESThe matching polynomial a (G; x) of graph G is a well studied corn-binatorial object", The collection of its roots, XI(m) (G), xz(m) (G), ... , XN(M) (G),where N is the number of vertices in G, defines the matching spectrum of G.The matching spectra of molecular graphs have found applications in thetheory of aromaticitys-". .The recursive formula=" for the evaluation of a (G; x) is of asimpIerstructure than the recursive formula" for the evaluation of ep (G; x), There-fore, the matching spectra exhibit an analogon of TEMO for wider classesof isomers than the MO energies. We present now such a class.Class 6, - The isomers of this class are bridged annulenes. Let N be aneven number, N = 2n. If the bridge is established between positions n andl we denote the corresponding graph by Br, l = 1, 2, ' , " n - 1. For l = 2n,we write A = BI. Graph A and a representative of the series of B/s aredepicted in Figure 7.TEMO IN SOME ISOMERS 5972n 2n2n-1~",\I l, /I III 1+1II II //n+1 n n-I n +'1 n n-tA BIFigure 7.By comparing the matching polynomial of A with the matching poly-nomial of any of BI's the following result is obtained:a (BI; x) - a (A; x) =: [a (PI-I; x)]2 ;::::O (9)where: a (Pk; x) = ep (Pk; x). Inequality (9) implies that the following ana-logon of TEMO (2):XI (m) (A) ;::::XI (m) (B) ;::::x2 (m) (B) ;::::x2 (m) (A) ;::::... ;::::(10)holds where B stands for any of Bi> l = 1, 2, ... , n - loAcknowledgement. - The work was supported in part by Self-ManagingAuthority for Science of Croatia, Yugoslavia, and also by the National ScienceFoundation (Grant No F6F 006Y).REFERENCE S1. O. E. P o lan sky and M. Z and e r, J. Mol. Struct. 84 (1982) 361.2. O. E. P o 1a n sky, J. Mol. Struci. 113 (1984) 281.3. A. G rao vac and O. E. P o lan sky, Croat. Chem. Acta 57 (1984) 1595.4. l. Motoc, J. N. Silvermann, O. E. Polansky, and Y. Olbrich,Theoret. Chim. Acta 67 (1985) 63.5. For the use of graph theory in chemistry see, e. g. A. G rao vac, I. G u t-m a n, and N. Tri naj s t i ć, Topological Approach to the Chemistry ofConjugated Molecules, Springer-Verlag, Berlin, 1977.6. I, G u trna n, A. G rao vac, and O. E. P o 1a n sky, Chem. Phys. Lett.116 (1985) 206.7. A. G rao vac, I. G u trna n, and O. E. P o 1a n sky, J. Chem. Soc., Fa-raday Trans. 2, 81(1985).8. C. D. God s i 1 and I. G u trna n, J. Graph Theory 5 (1981) 137.9. J. Ai h ara, J. Am. Chem. Soc. 98 (1976) 2750.10. I. G u t man, M. Mi 1u n, and N. Tri naj s t i ć, J. Am. Chem. Soc. 99(1977) 1692.598 J. HOXHA ET AL.SAŽETAKo nekim daljnim klasama izomera koji pokazuju topološki efekt namolekularne orbitaleJ. Hoxha, A. Graovac i O. E. PolanskyRazmatrane su neke daljnje klase izomera koji pokazuju topološki efekt namolekularne orbitale (TEMO). One se sastoje od dviju klasa heterosupstituiranihpoli ena i tri klase supstituiranih anulena. Šestu klasu čine premošteni anulenikod kojih TEMO vrijedi samo za spektre sparivanja.

On Some Further Classes of Isomers Which Exhibit Topological Effect on Molecular Orbitals

HRČAK - Portal of Croatian Scientific and Professional Journals

CROATICA CHEMICA ACTA CCACAA 59 (3) 591-598 (1986)CCA-1671YU ISSN 0011-1643UDe 541Original Scientific PaperOn Some Further Classes of Isomers Which ExhibitTopological Effect on Molecular Orbitals*Jahi HoxhaDepartment of Physics, Faculty of Science, University of Kosovo, Trg Maršala Titabb, 38000 Priština, Kosovo, Serbia, Yugoslavia,Ante GraovacRugjer Bošković Instite, P.O.B. 1016, 41001 Zagreb, Croatia, YugoslaviaandOskar E. PolanskyMax-Planck-Institut fur Strahlenchemie, Stiftstr. 34-46, D-4330 MULheim/Ruhr,F. R. GermanyReceived October 21, 1985Some further classes of isomers exibiting the TopologicalEffect on Molecular Orbitals (TEMO) are considered. They includetwo classes of hetero-substituted polyenes and three classes ofsubstituted annulenes. The sixth class contains bridged annuleneswhere TEMO is exhibited only for the related matching spectra.INTRODUCTIONThe regularity of the molecular orbital (MO) energies arrsmg form thedifferent topologies of isomers is called the »Topological Effect on MolecularOrbitals (TEMO)«1. The effect has been confirmed by experimental findings",as well as by quantum-chemical computational procedures of various levels ofsophisticationš-!The already existing cIasses of isomers which exhibit TEMO are extendedin the present paper by a few further cIasses. They incIude two cIasses ofhetero-substituted polyenes and three cIasses of substituted annulenes. Thesixth cIass contains bridged ann ul enes where TEMO is exhibited only forthe reI at ed matching spectra.The mathematics of TEMO is relatively simple. Different topologies ofthe isomers MA and MB under consideration are appropriately described bythe graphs A and B associated with the isomers", The corresponding charac-teristic polynomials ([> (A; x) and ([> (B; x) are then compared. There are vari-ous cIasses of isomerst= for which the difference function, il (B, A; x), satisfies:t.. (B, A; x) == ip (B; x) - ip (A; x) 2: O (1)* Presented at The International Symposium on Applications of MathematicalConcepts in Chemistry, Dubrovnik, September 2-5, 1985.592 J. HOXHA ET AL.The above inequality implies'A?X2k_l (A) 2:: x2k-l (B) 2:: x2k(B) 2:: x2k (A) 2:: .•. (2)where Xj (A) and Xj (B), j = 1, 2, ... , N, denote the roots of (JJ (A; x) and(JJ (B; x), respectively, i. e. the MO energies OI the isomers MA and MB as cal-culated within the simple MO theory. N stands for the number of conjugatedcenters in the considered isomer and equals the number of vertices in theassociated graph.The interlacing rule (2) is called TEMa! and it enables predictions onthe first absorption maxima, ionization potentials, etc., of the isomers MA andMB. In the case where the sign of ~ (B, A; x) generally changes along theenergy parameter x-asis, TEMa with inversion- holds.HETERO-SUBSTITUTED POL YENESA linear polyene, CH2(CH)N_2CH2,may be rep resen ted by the path graph,PN, with N vertices. Let us introduce two heteroatoms into the polyene atthe positions i and j. Let hi and h2 stand for the values of the Coulombintegral of the first and the second heteroatom, respectively. The correspond-ing (weighted) graph is denoted by (PN;i, j; hit h2)'Class 1. - Let N be an odd number, N = 2n-1. Let consider the hetero--substituted polyene MA with the first heteroatom at the position n in themiddle of the polyene and with the second heteroatom at the position 1,1> n. The related graph A = (P2n-1;n, 1;hit h2) is shown in Figure 1. Hereand in the further figures black a circle represents a heteroatom. Let usfurther consider the isomers of MA in which the first to heteroatom is movedout of the middle position to the new position 1 + P while the second hetero-atom remains at its position 1. These isomers are represented by Bp == (P2n-1;n + p, 1;hit h2), p = 1, 2, ... ,1- n-I, and a representative is de-picted in Figure 1.(}-D---O- _1 2 3hl---0---4.)--0- _n-l n n-]h2-0- --0l 2n-,cr-o--o- _2 3h·1--0----0---0--- -0- -0- __~n-l n n+' n+p 2n-lTEMO IN SOME ISOMERS 593The third factor on the right-hand side of eq. (3) is a nonnegative quantity.However, the parity of the second factor changes along x-axis and TEMOwith inversions holds. The inversion points are easily calculated after thevalues of 1, p and h2 are given and they don't depend on the length, 2n - 1,of the considered hetero-substituted polyenes.If the heteroatom h2 is moved to the left side, p < 0, of the middleposition, an equation analogous to eq. (3) can be derived by using the fol-lowing well-known identity:(4)The same comment applies to the rest of this paper as well.Class 2. - Let us now consider the situation where both heteroatomsare moved simultaneously by the same amount, p.Let N be an even number, N = 2n. Let us consider the hetero-substitutedisomer MA whose graph is A = (P2n; n, 1; ht, h2). Let us further consider theisomers M s, of MA which are represented by the graphs Bp = (P2n; n + p,1 + p; ht, h2). The difference function is rather complicated. In the special case1 = n + 1, i. e. where two heteroatoms are neighbours in MA and MBp , andfor both heteroatoms being of the same kind, hi = h2 = h, the differencefunction is simplified to the following form:(5)p = 1, 2, ... , n - 1The corresponding graphs are depicted in Figure 2.0---0---0-- _1 2 3h h• • --02n0---0- --0-- _1 2 3h h-0---0-___ _ • -0n n+l n s p n-p-t 2nFigure 2.Eq. (5) implies that TEMO with only one inversion point Xinversion = h,holds.In the case of electron attracting heteroatoms, h > 0, the inversion pointlies in the region of the bonding MO's. Up to the inversion point the regu-larity (2) holds where B stands for any of Bp's.In the case of heteroatoms with h < 0, the inversion point lies in theregion of the anti-bonding MO's. The regularity (2) holds in the whole regionof bonding MO's but the symbols A and B have to interchange their placesbecause !:J. (A, Bp; x) ::; ° for x ;?; o.594 ;T. HOXHA ET AL.SUBSTITUTED ANNULENESAn annulene, CNHN, may be represented by the cyc1e graph, CN, withN vertices.Further, we consider three c1asses of substituted annulenes.Class 3. - Let us represent the graph attached to the annulene CNHNby graph F. The linking site of the attached graph is represented by thevertex f. The graph obtained by deletion of f out from F is denoted by F- f.Let us consider a substituted annulene where two graphs FI and F2 areattached at the positions i and j, respectively. The corresponding graph isdenoted by (CN;i, j; r; F2)'Let N be an even number, N = 2n. Let us further take: FI = F2 = F. Letus consider the substituted annulene MB which is represented by graphB = (C2n; n, 2n; F, F) of Figure 3. If the graphs are attached at arbitrarypositions, a series of isomers MA, is obtained which are represented by graphsAl = (C2n; 1,2n; F, F), l = 1, 2, ... , n - 1. A representative of the series isdepicted in Figure 3.2n -1//(I/I,,,,,II/~n.l n n-lB=(C2n;n,2njF, F)Figure 3.The difference function is non-negative for the whole series:ep (B; x) - ep (Al; x) = 3 [ep (F - f; x»)2 • [ep (Pn-I; X)]2 > O (6)I = 1,2, ... ,n-lConsequently, TEMa with out inversions (2) applies if MO energies of anyof isomers MA, are compared with MO energies of the starting isom er MB.Let us note that B and Ar's together represent n isomers. By using theterminology of TEMOl Ar, A3, ••• , represent S-isomers of Model 23 as theypossess the planes of symmetry kr, k2' . .. . Similarly, A2, A4, ••• , representS-isomers of Model 33 as they possess the planes of symmetry O"r, 0"2, ••• , which•TEMO IN SOME ISOMERS 595pass through a pair of vertices. Altogether, there are (n - 1) S-isomers, [nI2][n-I]of them are of 1:-type and -2- of c-type. Brepresents T-isomer1 withregard to any of (n -1) S-isomers. The situation is c1arified in Figure 4.L2 )IIIIn~/n+J+l nAlA2BFigure 4.Class 4. - Let us consider substituted annulenes whose graphs are-shown in Figure 5. Let further N be an even number, N = 2n.2n-1-,,/IIII~.n+1 n n-1Figure 5.20Jt,,Ih\IHI//~.n+1 n n-1II~1t,t~The difference function for the isomer represented by B and for S isomerrepresented by some of Al'S is given by:1== 1,2,. o o, n-Iij) (B; x) - ij) (AI; x) ==h o ij) (F - f; x) o [ij) (pn-I; x)]2 (7)'596 J. HOXHA ET AL.-Consequently, TEMO with inversions holds where the inversion points arethe roots of ep (F - i, x) which is easy to calculate after the attached graphrepresented by F is specified.Class 5. - The isomers of this class can be understood as a special caseof Class 3 isomers. They contain two, generally different, heteroatoms withthe values hI and h2 associated with them. N is as before an even number,N = 2n. The isomers are rep resen ted by graphs which are depicted in Fi-:gure 6.2n2n-l~/ -' .•..•.:' \"2n2n-l~/' -'''',I \1! 91"~n+1 n n-1JI"I/.~n+1 n n-1Figure 6.'The difference function is easy to ca1culate and is given byp (B; x) - p (Al; x) = hI hz [P (Pn-'; x)]2 (8)I = 1, 2, ... , n-IAs a result, TEMO as is given by (2) holds if both hI and hz are of the sameparity. If hi and hz are of opposite parity, TEMO again holds but in (2) onehas to interchange the symbols B and A. Obviously, A stands for any of AI's.MATCHING SPECTRA OF BRIDGED ANNULENESThe matching polynomial a (G; x) of graph G is a well studied corn-binatorial object", The collection of its roots, XI(m) (G), xz(m) (G), ... , XN(M) (G),where N is the number of vertices in G, defines the matching spectrum of G.The matching spectra of molecular graphs have found applications in thetheory of aromaticitys-". .The recursive formula=" for the evaluation of a (G; x) is of asimpIerstructure than the recursive formula" for the evaluation of ep (G; x), There-fore, the matching spectra exhibit an analogon of TEMO for wider classesof isomers than the MO energies. We present now such a class.Class 6, - The isomers of this class are bridged annulenes. Let N be aneven number, N = 2n. If the bridge is established between positions n andl we denote the corresponding graph by Br, l = 1, 2, ' , " n - 1. For l = 2n,we write A = BI. Graph A and a representative of the series of B/s aredepicted in Figure 7.TEMO IN SOME ISOMERS 5972n 2n2n-1~",\I l, /I III 1+1II II //n+1 n n-I n +'1 n n-tA BIFigure 7.By comparing the matching polynomial of A with the matching poly-nomial of any of BI's the following result is obtained:a (BI; x) - a (A; x) =: [a (PI-I; x)]2 ;::::O (9)where: a (Pk; x) = ep (Pk; x). Inequality (9) implies that the following ana-logon of TEMO (2):XI (m) (A) ;::::XI (m) (B) ;::::x2 (m) (B) ;::::x2 (m) (A) ;::::... ;::::(10)holds where B stands for any of Bi> l = 1, 2, ... , n - loAcknowledgement. - The work was supported in part by Self-ManagingAuthority for Science of Croatia, Yugoslavia, and also by the National ScienceFoundation (Grant No F6F 006Y).REFERENCE S1. O. E. P o lan sky and M. Z and e r, J. Mol. Struct. 84 (1982) 361.2. O. E. P o 1a n sky, J. Mol. Struci. 113 (1984) 281.3. A. G rao vac and O. E. P o lan sky, Croat. Chem. Acta 57 (1984) 1595.4. l. Motoc, J. N. Silvermann, O. E. Polansky, and Y. Olbrich,Theoret. Chim. Acta 67 (1985) 63.5. For the use of graph theory in chemistry see, e. g. A. G rao vac, I. G u t-m a n, and N. Tri naj s t i ć, Topological Approach to the Chemistry ofConjugated Molecules, Springer-Verlag, Berlin, 1977.6. I, G u trna n, A. G rao vac, and O. E. P o 1a n sky, Chem. Phys. Lett.116 (1985) 206.7. A. G rao vac, I. G u trna n, and O. E. P o 1a n sky, J. Chem. Soc., Fa-raday Trans. 2, 81(1985).8. C. D. God s i 1 and I. G u trna n, J. Graph Theory 5 (1981) 137.9. J. Ai h ara, J. Am. Chem. Soc. 98 (1976) 2750.10. I. G u t man, M. Mi 1u n, and N. Tri naj s t i ć, J. Am. Chem. Soc. 99(1977) 1692.598 J. HOXHA ET AL.SAŽETAKo nekim daljnim klasama izomera koji pokazuju topološki efekt namolekularne orbitaleJ. Hoxha, A. Graovac i O. E. PolanskyRazmatrane su neke daljnje klase izomera koji pokazuju topološki efekt namolekularne orbitale (TEMO). One se sastoje od dviju klasa heterosupstituiranihpoli ena i tri klase supstituiranih anulena. Šestu klasu čine premošteni anulenikod kojih TEMO vrijedi samo za spektre sparivanja.

On Some Further Classes of Isomers Which Exhibit Topological Effect on Molecular Orbitals

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia

HRČAK - Portal of Croatian Scientific and Professional Journals