Lucas sequences and Fibonacci triads of graphs in PMO calculation on the charge-transfer bands of a...

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Lucas Sequences and Fibonacci Triadsof Graphs in PMO Calculation on theCharge-Transfer Bands of a Series ofEDA Complexes: Correlation withExperimental and AM1 Results

JAYANTA SARKAR,1 MANAS BANERJEE,2 ASOK K. MUKHERJEE2

1Okersa High School, Okersa, Burdwan-713 514, India2Department of Chemistry, University of Burdwan, Burdwan-713 104, India

Received 19 July 1996; revised 31 October 1996; accepted 7 November 1996

ABSTRACT: The use of Lucas sequences and Fibonacci triads of graphs in a completePMO calculation on a series of phenols is shown. The results were correlated withcharge-transfer absorption maxima of a series of molecular EDA complexes of p-chloranilwith these phenols as donors. The correlation brings out the PMO parameters in astraightforward way. The graph theoretical results are also shown to correlate well withthe donor ionization potentials calculated by the AM1 method, and this correlation yieldsa value of the sp2-C Coulomb integral, a, which agrees well with the one obtained fromphotoelectron spectroscopic results. Q 1997 John Wiley & Sons, Inc. Int J Quant Chem 63:817]825, 1997

Introduction

Ž . w xraph theory GT has found application 1 inG a wide variety of fields ranging from isomerw xenumeration 2, 3 to the study of conjugated

w xmolecules 4]7 and theoretical investigation ofw xfullerenes 8]13 . In such applications, the charac-

Ž .teristic polynomials CP of the graphs represent-ing the system under study are often the starting

*Correspondence to: A. K. Mukherjee.Contract grant sponsor: UGC, New Delhi, India.

points. Although quite a large number of methodsw xexist in the literature 14]16 . It is well known that

construction of CP even for a single graph ofmoderate size is extremely difficult and the taskbecomes more difficult if a family of graphs isrequired for a particular study. Balasubramanianw x w x17 and Barakat 18 developed computationallyfacile methods of construction of CP based on

w xsymmetric function theory, and Randic 19´showed how nicely the set of equations appearingin Barakat’s formulas can be written down using

wthe Young diagrams. However, such methods 17,x18 require calculation of traces of n powers of the

Q 1997 John Wiley & Sons, Inc. CCC 0020-7608 / 97 / 040817-09

SARKAR, BANERJEE, AND MUKHERJEE

adjacency matrix, where n is the number of ver-tices in the graph. This is impossible without theaid of a computer and the task is even moredifficult if a family of graphs is under study. The

w xmethod developed by El-Basil 20 through the useŽ . Ž .of Fibonacci triads FT and Lucas sequences LS

is extremely useful in this respect. However, aliterature survey shows that this method has foundvery little application in real chemical problems.

w xRecently, we demonstrated 21 the application ofthe method on HMO calculation on a series ofmethylbenzenes in a hyperconjugative model. Thisseries involved no fused rings and only FTs weresufficient. In the present article, we chose a seriesinvolving fused rings so that both FTs and LSs are

Ž .needed. A graph theoretical GT perturbationalŽ .molecular orbital PMO calculation was done in

this way on a series of phenols and, thereby, theŽ .charge-transfer CT transition energies of their

Ž .molecular electron donor acceptor EDA com-plexes with p-chloranil have been shown to have aregular trend as the number and position of thephenolic—OH group change in the donor moiety.

The FT]LS method builds up the graphic poly-nomial coefficients of the necessary graphs in asequential manner. Our object is to show that sincethe graphs and subgraphs needed for the PMOcalculation by the GT method on a family of com-pounds are included in such sequences, the methodis of great practical utility in streamlining PMOcalculations and that such calculations may beused to interpret the trends in CT transition ener-gies of EDA complexes.

EDA complexes were chosen in this study be-cause they have recently been reported to be po-tential materials with nonlinear optical propertiesw x22 . The present GT calculation was supple-

w xmented by an AM1 calculation 23 of the donorionization potentials which also are found to fol-low the same trend as predicted by the GT results.

Ž .Moreover, an estimate of the Coulomb integral aof the sp2-carbon atom, which is usually bypassedin HMO calculations by setting a s 0, is obtainedfrom this correlation.

Graphs Needed for the Present PMOCalculation

The systems under study are the p-type EDAŽ .complexes of p-chloranil acceptor with phenol,

p-quinol, resorcinol, a-naphthol, and b-naphtholŽ .donors . In addition, two other phenols, e.g., cate-

chol and phloroglucinol, were used in the series totest the correlation between the two theoreticalmethods—graph theoretical PMO and AM1. Ac-cording to the PMO method of Coluson and

w xLonguet-Higgins 24 , the unperturbed systemscorresponding to the above phenols are hydro-carbon radicals obtained by replacing the —OH by—CH . The relevant HOMO energies and eigenco-2efficients were calculated graph theoretically afterconstructing the CPs by the FT]LS method of

w xEl-Basil 20 . Graphs representing the unperturbedŽ .systems radicals are shown in Figure 1.

( )Calculation of p G, k Numbers and CPsof the Necessary Graphs Through theUse of Fibonacci Triads and LucasSequences

Ž . w xThe p G, k number 25 of a graph G is thenumber of k collections of nonadjacent k sub-2graphs of G. The CP of G is given in terms of thisnumber of

mk ny2 kŽ . Ž . Ž .p G ; x s y1 p G , k xÝ

ks0

Ž .c s nyqŽ . Ž .q y2 x , 1ÝsgS

FIGURE 1. Graphs for the unperturbed systemscorresponding to the phenols under study.

VOL. 63, NO. 4818

LUCAS SEQUENCES AND FIBONACCI TRIADS OF GRAPHS

where m is the maximum possible value of k in G;n, the total number of vertices in G; S, the collec-

Ž .tion of disjoint rings with q vertices q F n ; s, aŽ .member of the set S; and c s , the number of

Ž . Ž .components i.e., rings in s. Equation 1 is aw xvariant of the well-known Sachs theorem 26 .

Ž .A triad of graphs g , g , g is said to be a1 2 3Ž . ŽFibonacci triad FT if the set g y g , g y g ,3 2 3 1

. Žg y g consists only of the paths L and L i.e.,2 1 1 2.a single vertex and a path with two vertices . Here,

g y g means a graph resulting from pruning gi j jw xout of g 20, 25 . The following recurrent relationi

exists among the members of an FT:

Ž . Ž . Ž . Ž .p g , k q p g , k q 1 s p g , k q 1 . 21 2 3

� 4A sequence of graphs g , having a common sub-iŽ .graph g, is said to be a Lucas sequence LS if

� 4g y g , g y g , g y g , . . .1 2 3

� 4 Ž .s C , C , C , . . . , 3m mq1 mq2

Ž . w xwhere C means a cycle ring with r vertices 20 .rŽ .The p G, k numbers of the graphs in an LS obey

the recurrence relation

Ž . Ž . Ž . Ž .p g , k q p g , k q 1 s p g , k q 1 . 4m mq1 mq2

Ž . Ž . w xUsing relations 2 and 4 , El-Basil 20 developeda method for ‘‘building up’’ large graphs and

Ž .obtaining their p G, k numbers from smallergraphs. If a graph contains no fused ring, only FTsare sufficient; if fused rings are present in thegraph, both FTs and LSs are necessary. For the

Ž .second type, El-Basil defined a graph G i, j as oneresulting from the fusion of two linear chains Li

and L having i and j vertices, respectively, withja K from two sides as shown in Figure 2. Thus,2� Ž .4 � Ž . 4G i, j becomes an LS because G i, j y L is thei

� 4sequence C ; the same is true if i and j arejinterchanged.

Ž .For calculation of p G, k numbers of the graphsG ]G in Figure 1, which represent the hydrocar-1 5bon radicals corresponding to phenol, catechol,resorcinol, p-quinol, and phloroglucinol, respec-

Ž . Ž .tively, the FTs needed are FT 1 ]FT 5 as shown inŽ .Figure 3, and the recurrence relation 2 is used asŽ .follows: For the sake of simplicity, p k is written

Ž . Ž i.for p G, k and G is the starting graph of the0Ž .triad FT i :

FIGURE 2. LS generating fused rings.

FIGURE 3. The relevant FTs.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 819


Ž .Triad FT 1 :

Ž . Ž . Ž . Ž .p 0 p 1 p 2 p 3

Ž1.G 1 4 3 00o o o

C 1 6 9 26x x x

G 1 7 13 51

Considering the fact that the ring of G contributes17y6 Ž .y2 to the coefficient of x according to Eq. 1 ,

the CP of G can be written down immediately1from the third row of the above table:

Ž . 7 5 3 Ž . Ž .P G ; x s x y 7x q 13 x y 5 q 2 x . 51

w xThe Ulam subgraph 27 corresponding to thedeletion of the pendant vertex G is C ; it will be1 6required in the subsequent stages of the presentPMO calculation and so its CP is written downfrom the second row:

Ž . 6 4 2 Ž . Ž .P C ; x s x y 6 x q 9 x y 2 q 2 . 66

Ž .A similar calculation of p k numbers forŽ . Ž .FT 2 ]FT 4 is shown in Table I.Taking into account the ring contributions at

Ž .appropriate places, the CPs obtained from the p knumbers of Table I are

Ž . 8 6 4 Ž . 2P G ; x s x y 8 x q 18 x y 11 q 2 x q 12

Ž . 8 6 4 Ž . 2P G ; x s x y 8 x q 18 x y 10 q 2 x3

Ž . 8 6 4 Ž . 2P G ; x s x y 8 x q 18 x y 10 q 2 x q 1.4

Ž .7

The Ulam subgraph corresponding to the deletionof any one pendant vertex from all the three graphsG to G is G whose CP has already been found2 4 1

Ž .in 5 :

Ž .Triad FT 5 :

Ž . Ž . Ž . Ž . Ž .p 0 p 1 p 2 p 3 p 4

Ž5.G 1 6 8 0 —0o o o o

G 1 8 18 10 03x x x x

G 1 9 24 18 05

TABLE I( )Calculation of p k numbers for the triads

( ) ( )FT 2 ]FT 4 .

( ) ( ) ( ) ( ) ( )Triad Graph p 0 p 1 p 2 p 3 p 4

(2)( )FT 2 G 1 5 6 1 00o o o o

G 1 7 13 5 01x x x x

G 1 8 18 11 12(3)( )FT 3 G 1 5 5 0 00

o o o oG 1 7 13 5 01

x x x xG 1 8 18 10 03

(4)( )FT 4 G 1 5 5 1 00o o o o

G 1 7 13 5 01x x x x

G 1 8 18 10 13

Considering the ring contribution,

Ž . 9 7 5 Ž . 3 Ž .P G ; x s x y 9 x q 24 x y 18 q 2 x . 85

The Ulam subgraph of G is G whose CP was5 3Ž .already obtained in 7 .

For the remaining two graphs G and G in6 7Ž .Figure 1, we first require the p k numbers of the

Ž .naphthalene graph, G . By definition, G 4, 4 isNaphŽ .G . Using the LS recurrence relation 4 , theNaph

Ž . Ž .p k numbers of G 4, 4 were calculated and areŽ .shown in Table II. To arrive at G 4, 4 by El-Basil’s

‘‘building-up’’ procedure, the starting graphs areŽ . Ž . Ž .G 1, 1 , G 1, 2 , and G 2, 2 , shown in Figure 2.

Ž .Their p k numbers are obtained by direct count-ing; those of the others are ‘‘built up’’ as indicatedin Table II. To construct the CP of G fromNaph

Ž .these p k numbers, we require ring contributionswhich are shown in Figure 4. We then have

Ž . 10 8 6 Ž . 4P G ; x s x y 11 x q 41 x y 61 q 4 xNaph

Ž . 2 Ž . Ž .q 31 q 12 x y 3 q 6 . 9

Ž .Knowing the p k numbers of C , G , and G ,6 1 Naph

it is now easy to obtain those of graphs G and G6 7

of Figure 1 by the FT method. The family of

VOL. 63, NO. 4820


TABLE II( )Calculation of p k numbers for the naphthalene graph using LS.

( ) ( ) ( ) ( ) ( )Graph p 0 p 1 p 2 p 3 p 4

( )G 1, 2 1 6 6 0 —o o o o

( )G 2, 2 1 7 11 3 0o x o x o x

( )G 3, 2 1 8 17 9 0x x x x

( )G 4, 2 1 9 24 20 3( )G 1, 1 1 5 2 0 —

o o o o( )G 1, 2 1 6 6 0 0

o x o o x o x( )G 1, 3 1 7 11 2 0

x x x x( )G 1, 4 1 8 17 8 0

o o o oa( )g 2, 4 1 9 24 20 3

o x o x o x o x o( )G 3, 4 1 10 32 37 11 0

x x x x x( )G 4, 4 1 11 41 61 31 3

a ( ) ( )G 2, 4 = G 4, 2 .

graphs required for this purpose is shown in Fig-ure 5, from which the following two sets of FTscan be identified:

Ž . Ž . Ž .I. C , G , g , G , g , g , g , G , G ,6 1 1 1 1 2 2 Naph 6

Ž . Ž .II. G , G , g , g , G , G .1 2 3 3 Naph 7

Here, G and G correspond, respectively, to a-6 7Ž .and b-naphthol. Calculation of their p k numbers

Ž .from these FTs using Eq. 2 are shown in TablesIII and IV. Ring contributions are now to be added

Ž .to appropriate p k numbers to get the CPs of G6and G . Figure 6 shows the calculation of ring7contributions to the CP of G ; that of G is similar.6 7We thus have

Ž . 11 9 7 Ž . 4P G ; x s x y 12 x q 50 x y 87 q 4 x6

Ž . 3 Ž .q 58 q 14 x y 10 q 10 x

Ž . 11 9 7 Ž . 5P G ; x s x y 12 x q 50 x y 86 q 4 x7

Ž . 3 Ž . Ž .q 55 q 14 x y 9 q 8 x . 10

Graph-theoretical Calculation ofHOMO Energies of the Phenols by PMOMethod

w xAccording to the Coulson]Longuet-Higgins 24Ž .PMO method, the energy E of the highest occu-j

pied p-MO of a phenol is given by

0 2 Ž .E s E q h b C q 2 k y 1 b C C ,¨ ¨Ý Ýj j O r j C — O r j t jr r-t

Ž .11

where E0 is the HOMO energy of the unperturbedjŽsystem here, the hydrocarbon radical is obtained

¨ .by replacing —OH by —CH ; r, the location of2¨the —OH group; t, the location adjacent to r ; C ,r j

the coefficient of the r th atomic orbital in the jthMO; and h and k , the perturbation parame-¨ ¨O C — O

ters defined by

Ž .a s a q h b , b s k b . 12¨ ¨ ¨ ¨O O C — O C — O



FIGURE 4. Enumeration of ring contributions fornaphthalene graph.

Here, a and b are the standard Coulomb andresonance integrals for a benzene C atom and abenzene C—C linkage; a , the Coulomb integralOfor an oxygen atom contributing two p-electrons toconjugation; and b , the resonance integral of a¨C — O

¨C—O linkage in a phenol.All the p-MO energies of a hydrocarbon radical

can be obtained by finding the zeros of the corre-

FIGURE 5. Some graphs from which FTs can be sortedout.

TABLE III( )Calculation of p k numbers for the a-naphthol

graph by the FT method.

( ) ( ) ( ) ( )Graph p 0 p 1 p 2 p 3 ??? ??? ???

C 1 6 9 26o o o o

G 1 7 13 5 01o x o x o x o x o

g 1 8 19 14 2 01x x x x x

g 1 9 26 27 7 02o o o o o

aG 1 11 41 61 31 3Naphx x x x x

G 1 12 50 87 58 106

a ( )G = G 4, 4 obtained through LS.Naph

Ž .sponding P G; x . However, by the use of thew xBudan]Fourier series 28 , only the required eigen-

0 w xvalue, e.g., E , can be extracted out 29 . Thejquantities C 2 and C C can be obtained by usingr j r j t jthe equations

Ž .P G y v ; xr2 Ž .C s 13r j w Ž .xd P G ; x rdx xsx j

Ž .P G y e ; xr t Ž .C C s , 14r j t j w Ž .xd P G ; x rdx xsx j

where G y v and G y e mean, respectively, ther r tsubgraphs obtained by deleting from G the r th

Ž . w xvertex and the r, t edge 30]32 .

TABLE IV( )Calculation of p k numbers for b-naphthol

graph by the FT method.

( ) ( ) ( ) ( )Graph p 0 p 1 p 2 p 3 ??? ??? ???

G 1 7 13 5 01o o o o o

G 1 8 18 11 1 02x x x x x

g 1 9 25 24 6 03o o o o o

G 1 11 41 61 31 3Naphx x x x x

G 1 12 50 86 55 97

VOL. 63, NO. 4822


TABLE VWavelengths of charge-transfer absorption maxima for chloranil]phenol complexes in dioxan medium and

0 0 0 0the respective perturbational quantities; E = x b = y3.1x eV, where x is the jth eigenvalue of thej j j junperturbed system in PMO method with a as the zero and b as the unit of energy.

a b 0 2( ) ( ) ( )Donor I eV hn eV E eV Ý CD CT j r r j

Phenol 9.115 2.8398 0 0.5714Catechol 8.795 — 0.9144 0.5509p-Quinol 8.712 2.3414 0.9644 0.6646Resorcinol 9.126 2.8205 0 0.5833a-Naphthol 8.454 2.2564 0 0.4500b-Naphthol 8.640 2.3504 0 0.5294Phloroglucinol 9.157 — 0 0.6000

a [ ]Donor ionization potentials calculated by AM1 method 23 .b [ ]Experimental data have been taken from 38 .

Correlation of the Graph TheoreticallyObtained HOMO Energies withExperimental CT Transition Energiesof EDA Complexes

0 ŽValues of E with a as the zero level of en-j. 2ergy and C for the phenols calculated in ther j

above manner are shown in Table V, together withŽ .the experimental CT transition energies hn ofCT

the EDA complexes of p-chloranil with the phe-nols under study. Since with a given acceptor anda series of related donors hn is known to varyCT

wlinearly with the vertical ionization potentials 33,x34 of the donor, which, in turn, is the negative of

the HOMO energies of the latter, one expects

hn s yE0 y h ? b C 2¨ ÝCT j O r j

r

Ž . Ž .y 2 k y 1 b C C q Const., 15¨ ÝC — O r j t jr-t

where the constant term includes the electronŽ .affinity E of p-chloranil and a term D thatA

appears in the Mulliken]Ham]Platt equation

Ž .hn s I y E q D . 16CT D A

Ž 0.Figure 7 shows that hn q E bears an excellentCT j

linear correlation with Ý C 2 , the correlation coef-r r j

ficient being .96. The linear plot indicates that theŽ .term involving C C in Eq. 15 has no effect,r j t j

making k s 1. The slope of the line gives¨C — OŽh s 1.67 with b s y3.1 eV, obtained from theO

first four singlet]singlet absorption bands in.benzene . These values are in close agreement

Žwith Streitwieser’s recommended values h s 2,O

FIGURE 6. Enumeration of ring contributions for thea-naphthol graph.



Ž 0. 2FIGURE 7. Plot of hn q E against Ý C for theCT j r r jphenol series.

.k s 0.8 obtained from other kinds of experi-¨C — Ow xmental data 35 .

Correlation with AM1 IonizationPotentials

Ž .Since ionization potentials I of the donors areDequal to the negative of their respective HOMOenergies, one expects

I s yE s yE0 y h b C 2¨ ÝD j j O r j

r

Ž . Ž .y 2 k y 1 b C C y a. 17¨ ÝC — O r j t jr-t

The I values of the donors were calculated byDw xusing the MOPAC package 23 . Figure 8 shows

Ž 0. 2the plot of I q E against Ý C whose excel-D j r r jlent linearity immediately gives k s 1 and the¨C — Oslope, 5.7894, gives h s 1.87, which also is inOvery good agreement with Streitwieser’s recom-

w xmended value 35 . The intercept directly gives anestimate of the Coulomb integral, e.g., a s y5.8

Ž 0. 2FIGURE 8. Plot of I q E against Ý C for theD j r r jphenol series with I obtained by the AM1 method.D

eV, which is in excellent agreement with its value,y5.789 eV, obtained from photo-electron spectro-

w xscopic results 36, 37 .

CONCLUDING REMARKS

Figure 1 reveals that the subgraph obtained bydeletion of one pendant vertex of G is G , that of1 6G , G , and G is G , that of G is G , and that of2 3 4 1 5 3both G and G is G . Such subgraphs are6 7 Naphnecessary for the PMO calculation. We thus seethat if one arrives at the CPs of a family of graphsby the FT]LS method, one automatically passesthrough the subgraphs needed for calculation ofthe perturbational quantities involving the eigen-vector component C . In this way, the sequentialr j

nature of the CT]LS method gives an extra advan-tage in the graph-theoretical PMO calculation on afamily of compounds.

ACKNOWLEDGMENTS

Ž .The authors M. B. and A. K. M. thank theUGC, New Delhi, for financial support extendedthrough DSA. Valuable suggestions by the refereeare also gratefully acknowledged.

VOL. 63, NO. 4824


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