Indian Journal of ChemistryVol. 35A, February 1996, pp. 87-92

Papers

Unusual conjugation pattern in a class of benzenoid hydrocarbonsIvan Gutman" & Vesna Ivanov-Petrovic

Faculty of Science, University of Kragujevac, YU-34000 Kragujevac, Yugoslavia

andSergey Pogodin

Department of Organic Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Received 2 August 1995; accepted 29 September 1995

Cyclic conjugation in angulenes ( =unbranched catacondensed benzenoid hydrocarbons with onekink) has been examined. When moving along the hexagons of the angulene molecule cyclic conjuga-tion varies in a rather non-uniform manner. This pattern of cyclic conjugation is at variance withpredictions based on the analysis of Kekule structures, and differs from what has been observed inthe case of a number of other catacondensed benzenoid systems.

In 1977 a method was put forward, by means ofwhich one can determine the effect of individualcycles on the total zr-electron energy of polycyclicconjugated molecules 1·3. Since then, the theory ofcyclic conjugation, concerned with the energy-ef-fects of cycles in conjugated molecules, has beenelaborated in due detail+", This theory was appli-ed to a variety of zr-elecrron systems, amongwhich were numerous classes ofbenzenoid hydrocar-bons3•7•19. The vast empirical material collected+"!?made it possible to establish a few general regu-larities for the pattern of cyclic conjugation inbenzenoid systems. Of these, we re-state the fol-lowing two!",

(A) The energy-effects caused by cyclic conju-gation are almost always in harmony withthe predictions based on the examinationof Kekule structures. In particular, thenumber of times a cycle is conjugated inthe Kekule structures (in the sense of theconjugated circuit model":"), is roughlyproportional to the intensity of cyclic con-jugation, as measured by the respective en-ergy-effect.

(B) In @atacondensed benzenoid hydrocarbonsthe intensity of cyclic conjugation (as mea-sured by the energy-effect) is maximal atthe terminal hexagons, and diminisheswhen going towards the centre of the mole-cule. In linear hexagonal chains the energyeffects monotonically decrease from endtowards centre.

Rule (A) was found to hold for the great majorityof benzenoid systems, but a few violations of it

were observed8,15.19. Rule (B) was verified for po-, lyacenes II,benzo-annelated polyacenes 12, fibona-cenes 13and a few other homologous series con-taining chains of hexagons 15.

Violations of both rules (A) & (B) were recentlydiscovered in the case of phenes 19.

In this paper we show that the "anomalous"conjugation pattern found'? in phenes occurs in amore general class of benzenoid hydrocarbons,namely, angulenes, Angulenes are defined as un-branched catacondensed benzenoid systems-possessing a single angularly annelated hexagon(which is usually called a "kink"c3.c.\ see Fig. I). Ifthe hexagons are labelled consecutively by 1, 2, ... ,h and if the kink is at position k, then we speakabout the [h.kjangulene. The [h.klangulene will bedenoted by Ath.k). Its structure is depicted inFig. 1.

All members of the angulene series with sevenor fewer hexagons are known compounds". An-gulenes with h = 3,4 & 5 are typical stable poly-

A(h.k) ""- kink

Fig. I-The structure of the [h.k[angulene: conventionallv,~~k~(h+ I)/~ if h is odd and ~~k~h/2 if h is even:

phenes are angulenes for which k = (h + 1l':2 or k = h. 2

INDIAN J CHEM. SEe. A. FEBRUARY 1996

cyclic aromatic hydrocarbons. Angulenes withh = 6 & 7 are intensively coloured and highlyreactive chemical species.

In order to be able to envisage the unusualcharacter of the cyclic conjugation in anguleneswe first outline the results of a "classical" ap-proach to this problem, based on the study of Ke-kule structures2J·2/l·2'1.

Cyclic Conjugation in Angulenes as Deducedfrom Kekule Structures

We denote by K{GI the number of Kekulestructures of a conjugated system whose molecu-lar graph is G. A direct way to infer from the Ke-kule structures the extent of cyclic conjugation ofa particular cycle Z is the local aromaticity index,defined as23.26

LAI = LAI{G, Z] = 2 K{G-Zl/K{GIwhere G-Z stands for the subgraph obtained bydeleting from G the vertices of the cycle Z.The number of Kekule structures of the [h, k1 an-gulene is given by27K{A(h, kj] = k(h - k + 1)+ 1If Z is the i-th hexagon of A(h, k) then we have,

{

h - k + 1

KIA(h.k)-Z}= ~+ 1

for i= 1,2, ... , k-lfor i= kfor i=k+ 1, k+2, ... , h

from which it follows that the local aromaticity in-dex of the i-th hexagon of A(h,k) is equal to

!2(h-k+ 1)/[k(h-k+ 1)+ IJ" 2/[k(h-k+I)+ljLAIII;= 2(k+l)/[k(h-k+l)+lj for i = 1,2, ... , k - Ifor i = kfor i =k + 1, .... , h... (1)

Notice that according to Eq. (1) all hexagonswithin a linear segment of A(h,k) have equal LAI-values. Hence, from the analysis of the Kekulestructures we have' to conclude that within a line-ar segment of the angulene molecule, the extentof cyclic conjugation is everywhere (approximate-Iy) the same. This fact is illustrated in Fig. 2 withthe example of the (unreaiisticaIly large) isomersA(23,4), A(23,8) & A(23, 12)We define the quantity A.as,

A.=A(h,k)=[LAI(k -1)+ LAI(k + 1)j!LAI(k)." (2)

Substituting Eq. (2) back into Eq. (1), we obtainA(h.k)=h+1 ... (3)Hence, A is independent of the parameter k, i.e.,independent of the position of the kink. Further-

~- •••••••• A/23.12)

._ ••••••••• A/23.8)

0.3

0.2

0.1 .....---~-,.-. • • • • •• A/23.4)

5 10 15' 20

Fig. 2- Local aromaticity indices, Eq. (1), of the hexagons ofthe isomeric anaWeaes A(23,4), A(23,8) & A(23,12); the hex-agons are numbered as indicated in Fig. I; the curves forA(23,8) and A(23,12) are shifted upwards by 0.1 and 0.25,

respectively

more, A is a linearly increasing function of h, i.e.,it linearly increases with the size of the angulenemolecule.

Method of Calculation and ResultsThe method for the computation of the effect

of a cycle Z on the total n-e\ectron energy of aconjugated rr-electron system whose moleculargraph is G, has been described in due detail else-wherel-ldll.31. Employing HMO approximationand using j3-units, this effect is evaluated bymeans of the formula,

cf(G Z)=~ "'J In I ~(G, ix) I dx, n fI ~(G, ix) + 2~(G- Z, ix)1

in which ~(H, x) stands for the characteristic pol-ynomial of a graph H, and i = J - 1.

The [h.klangulene contains a total of h(h + ))12cycles: h hexagons, h - 1 ten-membered cycles,h-2 fourteen-membered cycles,h-3 eighteen-mem-bered cycles, etc. In Table 1 are presented the cf-values of all cycles of the four isomeric h = 9 an-gulenes; these may be considered as typical forthe angulene series:".

As seen from Table 1 the energy effects rapidlydecrease with the increasing size of the cycle andsoon become negligibly small (as far as their phy-sico-chemical significance is concerned). The way

GUTMAN et al.: UNUSUAL CONJ'UGATION OF BENZENOID HYDROCARBONS R9

Table I-The ef-values (in P.units) of all cycles of angulenes with 9 hexagons; the hexagons are labelled according to Fig. I;the larger cycles arc denoted so that, for instance, 2-5 indicates the Ill-membered cycle embracing the hexagons 2. 3. 4 & 5

Cycle Z Size ofZ Energy-effect of the cycle Z in the molecule G

G=A(9,2) G=A(9,3) G=A(9,4) G=A(9,S)

1 0 0.1804 0.1110 O.O1}52 0.09092 0 (l.O417 0.0854 0.0607 0.05323 0 0.0594 0.0354 O.OM2 0.04994 0 O.n457 O.OOOX 0.0347 0.06175 0 0.0425 0.0405 0.06119 0.03476 6 0.0420 0.0431 (l.O473 0.06177 6 0.0432 0.0435 (l.O450 0.0499X 0 0.0504 (>.0504 0.0509 0.0532I) 0 0.0905 (>.0902 0.01}0I O.OI)()1}

1-2 10 0.0170 0.0524 0.0335 0.02712-3 10 (>.001}0 0.0088 0.0277 0.01883-4 10 0.0164 0.0072 0.0071 0.02034-5 10 0.0118 0.0172 0.0068 0.00075-0 10 0.0106 0.0126 0.0180 0.00076-7 10 0.0109 0.0118 0.0142 0.02037-8 10 0.0135 0.0140 0.0153 0.01888-9 10 0.0242 0.0243 0.0249 0.02711-3 14 0.0060 0.0053 0.0217 0.01422-4 14 0.0034 0.0029 0.0028 0.01253-5 14 (UlO6!! 0.0023 0.0022 0.00224-0 14 0.0050 0.0076 0.0021 (I.()020

5-7 14 0.0047 0.0061 0.001}0 0.00220-8 14 (Ul057 0.0065 O.OOX4 (1.(1125

7-9 14 0.001}4 O.OOI}X 0.0 III (1.(1142

1-4 IX 0.0026 0.0021 0.0020 0.0 lOX2-5 IX (l.OO15 0.0012 (1.0012 0.1)0II3-6 IX O.OOJ7 0.0010 0.0009 0.00094-7 IX 0.0031 0.0047 0.0009 0.0001)5-8 IX 0.0033 0.0044 0.0066 0.00 II6-1} IX 0.0047 0.0055 o.oon O.OIOX1-5 22 0.0013 0:0010 0.0009 (>.00092-6 22 0.1)001) 0.0007 0.0006 0.00063-7 22 0.0020 0.0000 0.0005 0.00054..X 22 0.0025 O.OO3X 0.0006 0.00065-9 22 0.0030 0.0040 0.0060 0.0001}I-C) 26 O.OOOX 0.0006 0.0005 0.00052-7 26 0:0006 0.0005 0.0004 0.00043-X 26 0.0022 0.0005 0.0004 O.()()O44-9 26 0.0024 0.0035 0.1)005 0.00051-7 30 0.0006 0.0004 0.0004 0.00042-X 30 0.1)005 0.0004 0.0003 0.00033-9 30 0.0021 0.0004 0.0003 0.0004I-X 34 0.1)005 0.0004 0.0003 0.00032-9 34 O.l)()O5 0.0004 0.0003 0.00031-9 3X 0.0005 0.0003 o.OO() 3 0.0003

90 INDIAN J CHEM. SEC. A, FEBRUARY 1996

Table 2- The ef-values (in units of P) of the 6-, 10- & 14-membered cycles of A(17,4), A(17, 8), A(l8, 4) & A(l8, 8); the la-belling of cycles is same as in Table I

CycleZ SizeofZ Energy-effect of the cycle Z in the molecule G

G=A(17,4) G-A(17,7) G-A(18,4) G-A(18,7)I 6 0.0953 0.0903 0.0953 0.09032 6 0.0607 0.0506 0.0607 0.05063 6 0.0662 0.0437 0.0662 0.04374 6 0.0350 0.0434 0.0350 0.04345 6 0.0619 0.0469 0.0619 0.04696 6 0.0472 0.0608 0.0472 0.06087 6 0.0432 0.0355 0.0432 0.0355S 0 0.0422 0.0615 0.0422 0.06169 0 0.0420 0.0471 0.0421 0.0472

10 0 0.0421 0.0432 0.0421 0.0433II 0 0.0421 0.0422 0.0421 0.042212 0 0.0420 0.0419 0.0421 0.0420D 0 0.0419 0.041R 0.0420 0.041914 0 0.0420 0.0419 0.0419 0.041915 6 0.0434 0.0433 0.0420 0.042016 6 0.050R 0.0507 0.0434 0.043317 0 0.0912 0.0910 0.0508 0.050718 0 0.0912 0.0911

1-2 \() (l.O336 0.0245 0.0336 0.02452-3 \() 0.0277 0.0142 0.0277 0.01423-4 10 0.0072 0.0120 0.0072 0.01204-5 10 0.0009 0.0128 0.0069 0.01285-6 10 0.0175 0.0173 0.0175 0.01736-7 10 0.0123 0.0070 0.0123 0.00707-8 10 0.0105 0.0070 0.0105 0.0070R-9 10 0.0099 0.0173 0.0099 0.01749-10 10 0.0098 0.0123 0.0098 0.012310-11 10 0.0097 0.0105 0.0098 0.010511-\2 10 0.0097 0.0099 0.0098 0.009912-\3 10 0.0097 0.0098 0.0097 0.009813-14 10 0.0099 0.0099 0.0097 0.009714-15 10 0.0107 0.0107 0.0099 0.009915-16 10 0.0\36 0.0136 0.0\07 0.010716-17 10 0.0246 0.0245 0.0\37 0.0\3617-18 10 0.0246 0.02451-3 14 0.021R 0.0100 0.021R 0.01002-4 14 0.0029 0.0067 0.0029 0.00673-5 14 0.0022 0.0062 0.0022 0.00624-6 14 0.0021 0.0076 0.0021 0.00765-7 . 14 0.0071 0.0021 0.0071 0.00216-8 14 0.0049 0.0021 0.0049 0.00217-9 14 0.0039 0.0021 0.0039 0.0021R-IO 14 0.0036 0.0070 0.0036 0.00709-11 14 0.0035 0.0049 0.0035 0.004910-\2 14 0.0035 0.0040 0.0034 0.004011- \3 14 0.0035 0.0037 0.0034 0.003612-14 14 0.0036 0.0037 0.0035 0.003613-15 14 0.0041 0.0041 0.0030 0.003614-16 14 0.0054 0.0054 0.0041 0.004115-17 14 0.0045 (UI094 0.0055 0.005416-11l 14 0.0095 0.0094

GUTMAN et aL: UNUSUAL CONJUGATION OF BENZENOID HYDROCARBONS 91

A(23.12)

A(23.81

A(23.4)

5 10 15 20

Fig. 3-Energy-effects of the hexagons of the isomeric angu-lenes A(23,4), A(23,8) & A(23,12); the hexagons are num-bered as indicated in Fig. 1; the curves for A(23,8) andA(23,12) are shifted upwards by 0.01 and 0.03 ,8-units, re-

spectively

in which the ef-values of hexagons vary when go-ing along the angulene molecule, and their de-pendence on the position of the kink are illustrat-ed in Table 2 and Fig. 3. In order to make thecharacteristic features of these dependencies eas-ily comprehensible, the data given in Table 2 &Fig. 3 pertain to angulenes with unrealisticallylarge numbers of hexagons (h = 17 & 18, and 23,respectively); the behaviour of smaller angulenesis fully analogous-", In Table 2 are also given datafor 10- and 14-membered cycles, showing thatthey obey essentially the same rules as the hexa-gons, except that their energy effects are consider-ably smaller. The same conclusion applies to cy-cles of sizes 18 and greater; however, their ener-gy-effects are insignificantly small.

DiscussionThe most remarkable feature of cyclic conjuga-

tion in angulenes is that when going from one endof the molecule towards its centre, the energy-ef-fects of the hexagons first decrease, attain a mini-mum and then iricrease again, reaching a' newmaximum at the hexagon k - 1 (or at the hexagonk + 1, if we start from the other end). The mini-mum value of the energy effect is at the kink, adetail which is in agreement with the Kekulemodel. cf. Eq. (1). Essentially the same triple-min-imum behaviour is observed also in the case of

10-, 14-, 18-, ... -membered cycles, with the onlydifference that their energy effects are somewhatsmaller. These unusual forms of conjugation areeasily envisaged from Table 2 and Fig. 3.

In a number of cases it was found that cyclicconjugation is most pronounced around the endof a hexagonal chain and is minimal in its centralpart. This feature could be rationalized by a kindof (electrostatic?) repulsion between the aromaticsextets". In the case of angulenes, such an expla-nation would fail completely. Here it turns outthat the positions most favourable for the aromat-ic sextets are the hexagons 1, k - 1, k + 1 and h.Two of them (namely, the hexagons k - 1 andk + 1) are second neighbours, lying next to thekink. .

As pointed out above, deliberations based onKekule structures imply that the conjugation with-in each of the two linear segments of the angulenemolecule is more or less uniform. Our results in-dicate that this. conclusion may be a drastic over-simplification. In other words, in the case of angu-lenes the classical approaches to cyclic conjuga-tion20-23.26.28appear to be inadequate. _

The extent to which the Kekule-structure-baseddescription of cyclic conjugation agrees with themolecular-orbital-based ef(G,Z)-inodel is bestseen by comparing Figs 2 & 3. It may, at the firstglance, look as if the two approaches give similarpictures of cyclic conjugation around the kink, i.e.,in the part of the molecule embracing the hexag-ons k - 1, k & k + 1. However, a more detailedexamination reveals that this is not the case.

The LAI-values of the hexagons k -1, k &k + 1 conform to Eq. (3). If, in analogy to A(h,k)[Eqs (2) & (3)] we introduce the quantity A,A=A(h,k)=[ef(k-l)+ef(k+ l)]/ef(k) ... (4)

where ef(i) denotes the energy effect of the i-thhexagon of [h.kjangulene, then direct calculationshows that neither is A independent of k nor is ita linearly increasing function of h. From the datapresented in Table 3 it is evident that, in contrastto Eq. (3), A(h,k) varies with k, but is almost in-dependent of h.

On the basis of the results presented in this pa-per we conclude that the Kekule-structure-baseddescription of conjugation modes in polycyclicJl-electron systems is not always satisfactory. Or-ganic chemists should be warned that the Clararomatic sextet theory", the conjugated circuittheory20-22and the various modem versions of re-sonance theory23,26,2Rsometimes fail to recognizethe existence of certain subtle, but important cyc-lic-conjugation effects.

92 INDIAN J CHEM. SEe. A, FEBRUARY 1996

Table 3- The A-values of some [h, k]angulenes, Eq. (4)

h A(h,2) A(h,3) A(h,4) A(h,5) A(h,6) A(h,7) A(h,8) A(h,9) A(h,lO)

7 2.8840 2.0824 1.91598 2.8831 2.0694 1.84839 2.8753 2.0649 1.8314 1.778110 2.8777 2.0634 1.8276 1.756418 2.8777 2.0674 1.8300 1.7578 1.7337 1.7239 1.7239 1.725419 2.8768 2.0674 1.8300 1.7578 1.7337 1.7239 1.7254 1.7219 1.721920 2.8768 2.0674 1.8300 1.7578 1.7337 1.7239 1.7268 1.7233 1.7233

AcknowledgementOne author (IG) gratefully acknowledges the

support by the Lady Davis Fellowship Trust at theHebrew University of Jerusalem.

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33 Clar E, The aromatic sextet (Wiley, London), 1972.