Aromaticity in polycyclic conjugated hydrocarbon dianions

Journal of Molecular Structure (Theochem), 185 (1989) 249-274 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

AROMATICITY IN POLYCYCLIC HYDROCARBON DIANIONS

249

CONJUGATED

MILAN RANDIC Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311 and Ames Laboratory-DOE*, Iowa State University, Ames, IA 50011 (U.S.A.) DEJAN PLAVSIC

and NENAD TRINAJSTIC

The Rugjer BoikoviE Institute, (Received

P.O.B. 1016,410Ol Zagreb, Croatia (Yugoslavia)

11 July 1988)

ABSTRACT Dianions of polycyclic conjugated hydrocarbons are classified into aromatic, partially aromatic and anti-aromatic types. The classification has been based on a single structural concept: the conjugated circuits model. The negative charge in dianions is formally treated as a “double” bond contracted to a single carbon atom. Most dianions have been found to belong to the type “partially aromatic”, the degree of aromaticity being rather small, even zero or corresponding to weakly antiaromatic systems. The degree of aromaticity has been determined by the relative roles of RE (4n + 2 ) and RE (4n) which represent the positive and the negative contributions to molecular resonance energy, respectively. Our approach is contrasted to the still occasionally used peripheral model, the limitations of which are more serious than generally recognized. Its premise is that large (peripheral) conjugation is dominant, whilst in fact small conjugated cycles (circuits) play the dominant role in aromaticity and conjugation.

INTRODUCTION

The notion of aromaticity was introduced into chemistry very early [ 11, intimately associated with benzene as the prototype of an idealized aromatic system [ 21. Already Kekule considered associating the concept with structural molecular features which should serve for the characterization of such compounds [ 31. This should be contrasted with the equally early attempt of Erlenmayer to describe the concept of aromaticity by molecular properties [ 41. The early development of quantum chemistry brought important advances in the characterization of the aromaticity concept [ 51. The most significant result was the recognition of the role of (4n+ 2) and 4n K electrons in simple cyclic structures formulated as the now well-known Hiickel (4n+ 2) rule [ 61. An*Operated for the U.S. Department of Energy by Iowa State University under the contract W-7405-ENG-82. This work was supported in part by the Office of the Director.

0166-1280/89/$03.50

0 1989 Elsevier Science Publishers

B.V.

No.

250

other important step was made by Dewar [ 71 who defined the molecular resonance energy (RE ) as the departure of molecular binding energy from a simple bond additivity based on standard values for the contributing CC single and double bonds. More recently, Breslow [8] and Dewar [9] introduced the notion of anti-aromaticity which arises in cyclic compounds which are less stable than the reference acyclic structure. Despite these significant clarifications of the concept of aromaticity, the practical question of determining whether a compound is aromatic, and to what extent, remains unresolved for many polycyclic systems [lo]. Different authors typically use different criteria which has added to the confusion [ 111, even to some disenchantment as reflected in the occasional voice for abandoning the notion of aromaticity [ 121, its redefinition [ 131 or at least renaming it [ 141 so that it applies to a less diffuse class of structures. However, aromaticity as a concept, even if very elusive to a precise characterization, is so deeply entrenched in chemical thinking that its use refuses to abate [ 15,161, hence the notion of aromaticity provides a challenge to theoretical chemistry remaining one of the most enduring and tantalizing unclarified chemical concepts [ 5,171. The renewed interest in chemical graph theory [ 18-221 unexpectedly opened up a possibility for developing a very general theory of aromaticity in the framework of the conjugated circuits model [ 23-281. In this paper we will review the conjugated circuits model and describe its use for studying the conjugation and aromaticity in polycyclic conjugated hydrocarbon dianions. Even if one has limited interest in the compounds discussed here, the approach is so general that the discussion presented may serve as an introduction to the study of aromaticity in other conjugated systems, including heteroconjugated systems and their ions. We will also compare our model with other theoretical models of aromaticity such as the peripheral model [29], Hess-Schaad’s (HS) model [30] and the topological resonance energy (TRE) model [ 31-331, currently used to study conjugated dianions [ 34-361. THE CONJUGATED CIRCUITS MODEL

The bases of our model are Kekule valence structures which have been in use in chemistry for over 120 years [ 1,2,37]. In the early development of the valence bond (VB) method, Kekule valence structures offered the basis for the description of a molecule as a superposition of different valence forms [ 381. Important properties and limitations of such descriptions were recognized rather early by several people [ 39,401. Recent interest in the application of graph theory to conjugated systems has brought to light additional properties and restrictions concerning Kekule valence structures [ 20,411. For example, the connection of the valence structures with permutations has been elaborated by the Zagreb group [ 42,431, whilst at the same time inherent limitation of the concept of parity of Kekule structures, introduced by Dewar and Lon-

251

guet-Higgins [ 441, has been discerned: the pairwise compatibility of parity cannot be extended to systems constructed by fusion of more than two odd rings at the same carbon site [ 45-481. Kekule valence structures provide a “natural” language for the description of molecules and are better suited for the discussion of chemical phenomena than are a collection of molecular orbitals [ 491, which appear useful in discussion of spectral properties [ 501. Diminished use of Kekule valence structures has been primarily due to computational difficulties in obtaining quantitative results via the VB formalism [ 401. A bypass of the difficulty is an attempt to supplement such results by “translating” the more readily available MO results to VB language by defining a suitable correspondence [ 51,521. An illustration of such efforts is the Kekule index [ 53 1, which attributes to individual Kekule valence structures a numerical weighting parameter. Finally, Herndon [ 54-561 initiated a VB-type computation which has developed into a general method for calculating molecular properties which is comparable to the semiempirical SCF n-MO methods. Numerous graph-theoretical papers then emerged which are concerned with various aspects of the use of Kekule structures [ 16,20,40,57-641. The resurrection of Kekule valence forms as a viable tool for discussion of the chemistry of conjugated systems is becoming apparent [ 65-681. The renewed interest in Kekule valence structures has led to a very striking finding [ 23 ] : individual Kekule structures contain circuits in which there is a regular alternation of CC single and double bonds. Such circuits have been named conjugated circuits. The concept of conjugated circuits allows one to view polycyclic conjugated hydrocarbons, instead of a collection of Kekule valence forms, as a superposition of conjugated circuits contained in all its valence forms. For example, there are two symmetry unrelated Kekule valence structures for naphthalene

I II Each of the above KekulB structures can be examined separately focusing alternation to individual rings. Structure I consists of two Kekule formulae of benzene (fused across the double bond), whilst structure II has one Kekule benzene formula only. The Kekule benzene formulae give rise to conjugated circuits of size six, because they represent an alternating sequence of three CC single and three CC double bonds, respectively. The other ring in II does not give rise to a conjugated circuit of size six because it does not contain a sequence of alternating CC single and double bonds. However, bonds on the periphery of this ring coupled with bonds on the periphery of the other ring give rise to a conjugated circuit of size ten, because it represents a regular alternation of five CC single and five CC double bonds, respectively. The symbol R,

252

is used for circuits of size 4n+ 2 (n= 1,2,...). The conjugated-circuits of the Kekule structures I and II of naphthalene is given below.

content

1: 0

0

RI II:

RI

0 cl RI Since naphthalene possesses three valence structures, R2

the total decomposition of the conjugation into conjugated circuits is given by: 4R1+ 2Rz. Examination of a large number of polycyclic conjugated hydrocarbons has shown that conjugated circuits of various sizes can appear which can be grouped into those of type 4n+ 2 and those of type 4n (denoted by Q,). All benzenoid hydrocarbons, for which there is hardly any dispute that they represent typical aromatic systems [69,70], show only the presence of the 4n+ 2 conjugated circuits. This suggests that the aromaticity and aromatic character can be tied to the existence of such conjugated circuits in a structure [23-281. Kekule structures of an arbitrary conjugated hydrocarbon can generally be decomposed into linearly independent, linearly dependent and disjoint conjugated circuits. Linearly independent circuits are those that cannot be obtained by the superposition of conjugated circuits of smaller size. In the present work we will consider only linearly independent conjugated circuits. There are several very significant consequences of such a structural approach to the characterization of aromaticity. (i) Predictions of aromatic character can be made solely from information contained in the Kekule structures of a molecule. (ii) The Hiickel rule has a very natural extension to polycyclic conjugated systems: it is not the number of 4n+ 2 K electrons which is the critical concept, but the presence of 4n + 2 conjugated circuits in the collection of Kekule valence structures for a molecule. Application of this 4n+ 2 conjugated circuits rule to monocyclic systems coincides with the use of the 4n+ 2 Hiickel rule, which represents a straightforward reduction of the former. Finally, (iii) the simple enumeration of conjugated circuits for a conjugated system leads to a formula for molecular resonance energy (RE ), which is given by the average content of conjugated circuits in valence structures calculated by dividing the total number of conjugated circuits by the number of Kekule valence structures

RE=;C

(a,R,+bzQ,)

n,l

where K is the number of Kekule structures of the conjugated molecule, R, and

253

Q, are the parametric values for the conjugated circuits of size 4n + 2 and 4n, respectively, and a, and b, are the number of R, and Qn circuits belonging to a given polycyclic molecule. We will consider only conjugated circuits of size n = 3 or less. In doing this we follow the empirical findings that only the smallest circuits make appreciable contributions to the resonance energy [ 54-561. In order to compare molecules of different sizes, we will use the RE per electron, RE/e, which is defined as [ 301 RE/e = REIN

(2)

where N is the number of K electrons in the molecule. The application of the conjugated circuit model is of profound importance: not only does it lead to the clarification of ambiguities associated with the use and misuse of the aromaticity concept, but it has also opened new directions for studies of the chemistry of conjugated systems and clarified differences among compounds already studied. For example, Hafner and Fleischer [ 711 synthesized aceheptylene (III), a non-alternant conjugated hydrocarbon, which shows different properties from other non-alternant hydrocarbons such as acenaphtylene (IV) [72] or pentalene (V) [73].

111 IV V Initially it was by no means obvious why III is different from other nonalternants: however, it is now known that, in contrast to, for example, IV and V, III is a non-alternant system having both 4n+ 2 and 4n conjugated circuits ( 4R2 + 2Q3), whilst IV possesses only (4n + 2 ) conjugated circuits (4R, + 2Rz) and V only 4n conjugated circuits ( 2Q2). It should be noted that 4n conjugated circuits typify the destabilizing contributions to the RE. Thus, IV is predicted aromatic, V anti-aromatic and III non-aromatic. These predictions are supported by experimental data [ 71-741. The peripheral model, however, predicts [36] both III and IV to be nonaromatic. This prediction is not supported by the experimental evidence [ 361 which reveals III as a non-aromatic species and IV as an aromatic species. Analysis of the conjugation via the enumeration of conjugated circuits in the collection of Kekule valence forms for a structure can frequently answer important questions even without necessarily resorting to some numerical parametrization for the graph-theoretical quantities R, and Q, (n= 1,2,...). All that is required is some assumption on the relative magnitudes of the trend. Since it is generally known that benzene is more aromatic than higher homologs, e.g. (4n+ 2) annulenes [ 751, it is plausible to assume that the relative

254

importance of conjugated circuits decreases with ring size, i.e. in general it will hold that R, > R2 > R3 > R, etc. This assumption alone implies that phenanthrene with the normalized decomposition ( 10Rl + 4R2 + R,)/5 will have a higher RE than anthracene with the decomposition ( 6R1+ 4Rz + 2R3) /4. Observe that quantum chemical calculations can only answer the question: “Which, anthracene or phenanthrene, will have a greater RE?” after the computations have been made, whilst the graph-theoretical analysis can be used to produce the answer, in many cases, on the back of the envelope. However, in order to obtain the numerical values from graph-theoretical analysis, outside information (e.g. information on the REs of selected standard compounds) must be used. Hence the two methodologies are complementary, not competitive! The numerical values of the parameters R, and Q, are obtained from applying the parametrization procedure to Dewar’s SCF n-MO resonance energies [ 761. Values of the parameters obtained [ 23,281 are as follows: RI = 0.869 eV; R,=0.247eV;R,=0.100eV;Q,=-0.781eV;Q,=-0.222eV;Q,=-0.090eV. CONJUGATED CIRCUITS IN DIANIONS

In this section the extension of the analysis of conjugation in dianions on naphthalene dianion is outlined (see Fig. 1). In all there are 11 non-equivalent ionic valence structures that must be analyzed: they have symmetry weighting factors of 4,2 or 1. First, it must be clarified how the negative charge participates in conjugation. The structure of cyclopentadienyl anion (VI) suggests that the negative charge should be formally counted as a double bond, as this makes apparent the similarity to benzene.

0I\ VI Strictly speaking the conjugated circuit R, in C.&H; has a negative charge, but, as discussed elsewhere [ 771, the minor difference between RI and R 1 can be neglected. Hence, throughout this discussion no differentiation will be made between conjugated circuits involving charge and those not having charge. The prime difference between different conjugated circuits is whether they are of the R type or the Q type. Each of the 11 valence structures in Fig. 1 are composed of contributions from the conjugated circuits as shown under the structure. The first four structures in Fig. 1 have a benzene Kekule ring RI, the other ring being part of the 12 n-electron periphery of the naphthalene dianion or the 8 n-electron periphery of the single charged ring. The next three structures have only a single conjugated circuit: the periphery of a doubly charged naphthalene. The next three structures have 8 a-electron rings and 12 n-electron

255

i(R,

4(R1+Q3)

\/ 00

+&I

4( R, +Q$

I ,/,03

I ‘I 03

2iR,+Q21

2(Qj

2(Q,I

4(Q,

4(Q3)

+Q,I

2(Qz+Q,I

4(Q, +a,)

(2Qz+Q$

Fig. 1. Non-equivalent KekultStype valence structures for a dianion of naphthalene and the count of the conjugated circuits. The negative charge is formally equivalent to a double bond, i.e. it counts as a pair of coupled Ir-electrons located on an atom instead of being distributed along a CC bond.

peripheries, and, finally, the last structure has two 8 n-electron rings (fused across the central CC bond) and a 12 n-electron conjugated circuit of the naphthalene periphery. By adding the contributions from all 31 valence structures the RE of the naphthalene dianion is obtained as RE = (12R,

+

18Q2 + 25Q,) /31

(3)

The dianion has both aromatic and anti-aromatic contributions; each component being appreciable. Using numerical values for the parameters R, and Qn (previously adjusted for neutral hydrocarbons) [23,28] it is found that RE(4n+2)=RE(12RI/31)=0.336 eV and RE(4n)=RE(18Q,+25QS)/ 31= - 0.201 eV. The molecular RE of naphthalene dianion is then equal to 0.135 eV. Note that the RE of the parent hydrocarbon is 1.323 eV. Thus, the naphthalene dianion retains only a small fraction ( 10% ) of the aromatic stabilization of naphthalene and is essentially non-aromatic.

256 DEGREE OF AROMATICITY

The degree of aromaticity A

RE(4n+2)+RE(4n) =RE(4n+2)-RE(4n)

(or anti-aromaticity

) is defined as [ 781 (4)

or AA_RE(4n)+RE(4n+2) -RE(4n)-RE(4n+2)

(5)

where RE (4n+ 2) and RE (4n) are the contributions to the molecular resonance energy arising from the 4n + 2 and 4n conjugated circuits, respectively. The degree of aromaticity (anti-aromaticity) depends on whether the contributions from 4n + 2 (4n) dominate, i.e. if the outcome is a positive (negative) RE. If the contribution from the 4n+2 and 4n conjugated circuits is balanced, then such compounds may be considered non-aromatic species. BENZENOID DIANIONS

The REs and the degrees of aromaticity, A, for a number of known benzenoid dianions (see Fig. 2) are listed in Table 1. If the RE/e and A indices for the compounds listed in Table 1 are compared with those of benzene dianion, it is seen that benzene dianion is rather atypical. In fact, benzene dianion would be a prototype of anti-aromatic benzenoid dianions having only Q2 (a electron conjugated circuits of size 8). Naphthalene becomes nearly zero-aromatic in view of the almost complete cancellation of the aromatic and anti-aromatic contributions. As the size of benzenoid dianions increases their RE and their aromatic character increases, as indicated by A. On comparison of RE and A values for isomers, like anthracene and phenanthrene, a trend can be discerned in that systems which are more benzenoid (phenanthrene) and have a larger RE as a neutral molecule still have a larger RE as dianion, but also contributions of anti-aromatic 4n conjugated circuits increase thus making the A index smaller. In other words, molecules that are considered more aromatic (i.e. the benzenoid character is more pronounced) have more to lose and have a proportionally lower aromatic content as dianions. Another minor point is the small difference in RE of the dianions of chrysene and benzphenanthrene which as neutral systems have the same RE; there is even a difference in the number of Kekule valence forms for the two compounds (189 vs. 188). This difference can be attributed to the different number of so-called “long” bond structures that one can write for the two isomers (117 vs. 116) and this can be verified from the corresponding coefficient of the Wheland polynomials for the structures [ 791.

251

03;

aD=

09; IX

Fig. 2. Structures of the benzenoid dianions studied.

TABLE 1 Molecular resonance energies (BB), numerical values of the resonance energies and the degrees of aromaticity for dianions of benxenoid hydrocarbons Benxenoid dianion

RE expression

BE

BE/e

(eW

(eW

0.135 0.541 0.580 0.899 0.995 1.035 1.048 1.151 0.839 1.486 1.573 1.454

0.011 0.034 0.036 0.045 0.050 0.052 0.052 0.058 0.047 0.062 0.066 0.066

A

Bef.

0.251 0.662 0.621 0.825 0.784 0.757 0.760 0.780 0.748 0.837 0.849 0.841

101 101-105 101,104 101,102,104,105 101,104 101,104 101,104 101,104 101,105-107 101,104 101,104 101,102,105

258

In comparing molecules of different size, such as naphthalene, anthracene, tetracene, pentacene, etc., or phenanthrene, chrysene, picene, fulminene, etc., which illustrate families of cata-fused benzenoid systems, it is found that while the aromatic characteristics of such molecules decrease with the size of the compound, the aromatic characteristics of their dianions increase with size. The aromaticity here is only partial because the delocalization associated with the 4n + 2 conjugated circuits is opposed by contributions arising from the 4n conjugated circuits. However, as the size of the dianion increases the relative contribution of the anti-aromatic components diminishes. This has an interesting possible consequence: the stability of graphite lattice will not be drastically affected by charge, in general, and charge when present will spread locally to become distributed over a greater fragment, thereby increasing the RE of the system. Some benzenoid dianions, e.g. pyrene dianion, that are predicted to be aromatic [e.g. A (pyrene dianion) = 0.751, are predicted by the peripheral model to be anti-aromatic [80]. Other theoretical models (e.g. the HS model [35] ) and experimental data [81] indicate that pyrene dianion is aromatic but to a much lesser degree than the parent hydrocarbon. NON-BENZENOID ALTERNANT DIANIONS

Non-benzenoid alternant systems are composed of fused even-sized rings; examples being biphenylene and octalene. A number of such molecular skeletons have been analyzed from the graph-theoretical point of view; typically the analyses have shown a neutral molecule the RE of which is comprised of both (4n+ 2) and 4n conjugated circuit contributions. In the case of biphenylene, the RE was found to be only 0.360 eV (as compared with phenanthrene which has the same number of Kekule valence structures and has an RE of 1.955 eV, a value more than five times greater). The expression for the RE of biphenylene [ 251 is (8& + 2Q, + 4QB+ Q3)/5 which shows the source of the reduced value of the RE is due to the partial cancelling of the positive 4n+ 2 contributions with negative 4n contributions: 1.390 eV and - 0.508 eV, respectively. The index of partial aromaticity is therefore (1.390 - 0.508) / (1.390 +0.508) = 0.465 or 46.5%. The results of the enumeration of the conjugated circuits of dianions (see Fig. 3 ) are shown in Table 2. The character of the neutral and dianion systems has not changed in type, only in degree. For example, biphenylene dianion has lost some partial aromaticity, numerically from 46.5% to about 40%. On the other hand, non-aromatic benzocyclobutadiene (having predominantly 4n components, RE - 0.089 eV, with a low value of the anti-aromaticity in(%+2Q,+2Q,)/3= dex of 7% [ 281) as dianion is still non-aromatic but the predominance of antiaromatic 4n conjugated circuits is now shifted to 4n+2 circuits (A = 16%). The limited experimental data does not allow much elaboration, but the find-

259

o=a=co’ w= cD=am= cco; xx

XIX

XXIII

XXI

XXIV

XXII

XXV

cm=cm= a= XXVI

XXVII

XXVIII

=

OB XXIX Fig. 3. Structures of the non-henzenoid alternant dianions studied. TABLE 2 Molecular resonance energies, numerical values of the resonance energies and the degrees of aromaticity for dianions of non-benzenoid altemant hydrocarbons

hf.

Non-benzenoid alternant dianion

RE expression

RE (eV)

RE/e (eV)

A

XIX

(16R*)/16 (6R,+16R,+6Q,+9Qz)/20 (12R1+16Rz+36R3+15Q,)/45 (40R1+32R2+32R3+12Q1+48Q,)/60 (32R,+24Q,)/61 (80R1+16RZ+72R2+48Q2+12Q3)/108

0.246 0.123 0.325 0.431 0.042 0.638 0.142 0.430 0.212 0.669 0.718

0.025 0.012 0.023 0.031 0.003 0.036 0.007 0.024 0.013 0.037 0.040

1.000108-113 0.156 0.687101,114 0.393101,115,116 0.193 117 0.746 118,119 0.325101,120 0.651 101,114 0.245 0.756 0.848

xx XXI XXII XXIII XXIV xxv XXVI XXVII XXVIII XXIX

(24RI+32R,+72R,+73Q,+24Q,)/124 (50R1+28R2+36R3+40Qz+28Q3)/99 (36R,+54R,+18Q,+49Q,+23Qa)/83

(84R1+18Rz+72R3+48Q2+12Q,)/109 (94R,+18R2+78R3+30Q2+12Qa)/120

ings from Tables 1 and 2 may perhaps be summarized as: dianions of benzenoid and alternant non-benzenoid systems seem to show a trend in that the presence of the negative charge gives a consequent decrease in the aromaticity or partial aromaticity for neutral systems classified as aromatic or predominantly

260

aromatic, while a negative charge in non-benzenoid systems classified as neutral or anti-aromatic (i.e. a predominantly anti-aromatic component) has the effect of diminishing the role of the anti-aromatic components. The number of available non-benzenoid dianions is, however, too small for more firm conclusions. Nevertheless, it can be seen how the concept of conjugated circuits offers a capability for the quantitative characterization of molecular properties that have been hitherto describable only in very qualitative terms. For instance, studied dicyclo-octatePaquette and co-workers WI traeno ( 1,2 : 4,5)benzene dianion (XXV) and described the molecule as a planar 20 n-electron structure, where the electronic repulsion is adequate to overcome the strain energy and the anti-aromatic character of the ion. This description is fully corroborated by our theoretical analysis: the RE of XXV is given by: ( 24R1+ 32Rz + 72R3+ 738, + 24Q,)/124 in which the positive aromatic contribution is almost cancelled by a large negative anti-aromatic contribution resulting in the rather small RE value (0.142 eV) . The aromatic index of XXV is 32%, indicating a slight aromatic dominance. NON-ALTERNANT

NON-BENZENOID DIANIONS

The real test of the aromaticity concept and criteria are non-alternant nonbenzenoid conjugated systems. Here, neutral molecules already offer grounds for disagreement, and contradicting claims have been made in the past with respect to many members of this class, some members of which show properties that invite a classification as aromatic (e.g. azulene), while others have novel complexity as exhibited for the first time with Hafner’s hydrocarbon (a compound which shows strong differences from usual aromatics). It was only on the application of graph-theoretical analysis to such molecules that it was realized for the first time that non-alternant conjugated hydrocarbons also form two fundamentally different classes (as is the case with alternants benzenoid and non-benzenoid, e.g. phenanthrene and biphenylene, respectively) [ 251. In analogy with benzenoid and non-benzenoid classification these non-alternants can be called azulenoid and non-azulenoid, azulene being the most well known and simple representative of non-alternant systems having only conjugated circuits of size 4n + 2. Table 3 gives the results of the enumeration of the conjugated circuits for azulenoid and non-azulenoid dianions, the structures of which are shown in Fig. 4. Besides the RE and the A index for each dianion, Table 3 gives the A index for a neutral molecule in order to facilitate comparisons. A number of interesting results can be observed: for example, the striking difference in the behaviour of the pentalene and heptalene dianions. As neutral molecules, both are anti-aromatic systems, pentalene characterized by Q2and heptalene by Q3, conjugated circuits, the difference being in degree but not in kind. When dianions are examined, pentalene, a prototype of anti-aromatic systems, be-

(ZORi+16R2)/20 (15R,+14Q,+25Q,)/31 (36R,+42Qz)/45 (4%R,+36R2+36R3)/52 (6%R,+36R2+32R3)/56 (44R,+40R,+1%Q,+36Qs)/56 (25R1+12R2+45R3+56Q2+52Qz)/76 (9OR,+lO%R,+5%R,+%Q,+6Q~))/9O (l10R1+10%R2+96R3+1%Q2+52Q~)/102 (46R1+4%Rz+100R3+70Q2+100Q3)/123 (166R1+90R2+44R3+63Q2+57Q3)/139 (60R1+60Rz+2R,+50Qz+110Q3)/111 (150R,+ 106Rz+64R3+44Q2+ 12Q,)/133 (164R1+116R2+40R3)/123

xxx

xxx1 xxx11 xxx111 XXXIV xxxv XXXVI XXXVII XXXVIII XXXIX XL XL1 XL11 XL111

RE expression

Non-alternant dianion

1.066 0.24% -0.127 1.042 1.271 0.729 0.159 1.203 1.207 0.303 1.091 0.415 1.143 1.423

RE (eV)

0.107 0.021 - 0.009 0.074 0.091 0.052 0.010 0.075 0.075 0.017 0.061 0.023 0.064 0.079

RE/e (eV)

-

-

1.000 1.000 1.000 1.000 1.000 0.692 1.000 0.761 0.847 1.000 0.661 1.000 1.000

- 1.000

Parent molecule

A

1.000 0.41% 0.443 1.000 1.000 0.73% 0.261 0.959 0.876 0.431 0.799 0.523 0.875 1.000

Dianion 110,121,122 101 101,122 101,110,123,124 101,110,123 101,102,125 125,126 101,123 101,122,123,127 106 101,102 106 101,128 129

Ref.

Molecular resonance energies, numerical values of the resonance energies and the degrees of aromaticity for non-alternant dianions

TABLE 3

= 03 = a= co

262 = 03 xxx =

XXXIV

=

q

63

83

XXYIII

xxx11

xxx1

xxxv

=

& @ XXxvlI

XXXVI

= 43 = w =

=

=

0

@

XXXVIII

XXXIX

@

XL

XL1

=

%

XL111

XL11

Fig. 4. Structures of the non-alternant dianions studied.

comes fully aromatic, its conjugation being decomposed solely into RI and R, (singly and doubly charged) conjugated circuits. In contrast, heptalene has contributions from R3 and Q2, i.e. aromatic and anti-aromatic conjugated circuits, and is therefore of an intermediate class. The difference stems from the presence of 4n + 1 and 4n + 3 rings in heptalene: when an electron is added the rings convert to 4n + 2 and 4n, thus making the difference between the pentalene and heptalene dianions fundamental. Conversely, if we were to consider cations, abstraction of a single K electron would convert the two rings in 4n and 4n + 2, respectively, and we can therefore predict complementarity in character: pentalene dication is expected not to be aromatic or only partially so (if it is to be stable at all), while heptalene dication is expected to show aromatic characteristics.

0

W

XLIV XLV S-Indacene XLIV and its asymmetrical

isomer XLV as neutral

molecules

are classified as anti-aromatic, their conjugation is represented by Q3 antiaromatic conjugated circuits only. As dianions the two isomers both become fully aromatic, even though not fully equivalent in their conjugated-circuit counts. Most of the dianions in Table 3 are again of an intermediate type showing a variable degree of aromatic 4n + 2 and anti-aromatic 4n contributions. Acenaphthylene as a neutral system has the same conjugation content as naphthalene when conjugated-circuit counts are made, because the peripheral CC double bond does not participate in the overall conjugation. One way of acknowledging this is by recognizing the exocyclic bonds to naphthalene as essentially single (i.e. single in all Kekule valence structures); the other way of recognizing the deficiency of full conjugation in acenaphthylene is by recognizing that an external double bond is never involved in conjugation with others via conjugated circuits. When the corresponding dianions are compared, a substantial increase is seen in the aromaticity of acenaphthylene dianion (A = 74% ) compared to the hypothetical napthalene dianion (A = 25% ). Hence, restricted delocalization in neutral acenaphthylene confined to the naphthalene moiety is partly relieved in the dianion, but at the same time an increase in anti-aromatic composition occurs, due to the presence of QZand Q3 conjugated circuits, with the net result of some 74% aromatic character, compared to 100% in the neutral molecule. The results shown in Table 3 can be summarized by an observation which is consistent with the already mentioned qualities of 4n+ 1 and 4n+ 3 rings, the former contributing to the occurrence of 4n + 2 conjugated circuits while the latter make 4n conjugated circuits when additional charge is present. Thus, in general, pentalene and systems having a pentalene nucleus of a five-membered ring(s) have more aromatic content, as measured by the index A, in the dianions than in the neutral systems, while the role of the seven-membered rings is the opposite, i.e. they increase the degree of anti-aromaticity in dianions. Consequently, XXXVII, having a pentalene nucleus, is expected to show a high degree of aromaticity (A = 95.9% ) even though the parent neutral system is anti-aromatic (AA = 100% ) . The perycyclene dianion (XXXVIII) is also highly aromatic with A = 87.6%, which is higher than the index for the neutral parent molecule (A = 76.1% ). On the other hand, azupyrene dianion (XxX1X) has A =43.1% which is smaller than the corresponding aromaticity measure of neutral azupyrene (A =84.7%). The opposing contributions of the five- and seven-membered rings are partly responsible for the smaller net effect. A similar situation is again found in XL1 which as dianion has A = 52%, while the neutral molecule has a larger aromaticity content (A = 66.1% ) . The five-membered ring appears to make the dominant contribution, as compared with the seven-membered ring, as judged from the above examples and as well illustrated by azulene itself. Interestingly, enumeration of the conjugated circuits in naphthalene dianion and azulene dianion gives the same count of circuits (15R, + 14Qz+ 25&,), the difference being in the number of

264

Kekule valence structures which are 29 and 31, respectively, and which result in azulene having a fractionally smaller RE. The peripheral model of Platt, for example, leads to the prediction [ 361 that the acenaphthylene dianion (XXXV) is an anti-aromatic species, whilst the aceptylene dianion (XXXVI) is a non-aromatic species. However, other theoretical models [ 351, including the conjugated-circuits model of aromaticity, and the experimental data [ 361 show unequivocally that XXXV is a non-aromatic dianion, whilst XXXVI is an aromatic dianion. COMPARISON WITH OTHER METHODS FOR COMPUTING

RE

The results for a selection of the studied dianions for which computations have been reported using the so-called topological RE (TRE) are given in Table 4. The essential novelty of this approach is the definition of the reference structure [31-331. TRE measures the departure of the computed RE (and here, in principle, one can use a MO scheme of any type, starting with a simple HMO or, alternatively, adopting one of the SCF MO schemes available) from a standard, a hypothetical abstract system in which contributions from cyclic terms to the secular equation are eliminated. TRE is thus a graph-theoretical concept and it is of interest to see how such a scheme compares with our approach which is also basically graph theoretical. The particular results of 11% and Trinajstic [34] were based on the HMO, and thus the comparison will TABLE 4 Comparison between the REs derived in this work and the TREs for some conjugated dianions Compound XIX xx XXI XXII XXIV XXV XXVI XXVII xxx xxx111 XXIV xxxv XXXVIII XL11 XL111 “This work. bR.ef. 34.

RE” (eV)

TREb

0.246 0.123 0.325 0.431 0.638 0.142 0.430 0.212 1.066 1.042 1.271 0.729 1.207 1.143 1.423

0.186 - 0.045 0.193 - 0.085 0.228 - 0.039 0.189 0.030 0.464 0.475 0.573 0.294 0.539 0.516 - 0.033

(B)

show how Hiickel-type calculations fare in these cases. Given a proper reference structure, the Hiickel MO method has been shown to offer satisfactory results for predicting aromatic stability, even for heterocyclic systems [ 83-851. A glance at Table 4 (and Fig. 5) immediately reveals a good qualitative agreement between our numerical results (in eV) and the numerical predictions (in /3 units) of the TRE model. Dianions that we found to have a small RE have a small RE in the TRE approach also, and those that we found to have a relatively large positive RE also have a relatively large positive RE in the TRE approach, with the exception of two cases: XXII and XLIII. We believe that the cause of the disagreement in the case of XXII and XL111 is the numerical error in the computation of the TRE values [ 861. This conclusion is based on the otherwise good agreement between our RE values and those reported by Bates et al. [ 351 for XXII and XL111 (see Table 5). In general, there is a parallel between the TRE model and the conjugatedcircuits model. However, the approach based on the conjugated circuits is computationally much simpler than the TRE approach [ 66,87-901. In Table 5 we have compared our results with those of Bates et al. [ 351 who have reported resonance energies per atom (REPA) of a number of polycyclic conjugated dianions. We decided to compare the REs normalized in two ways with REPA indices of those dianions that have been studied by both groups. These are: resonance energies per z electron (RE/e) and resonance energies per atom (RE/a). All three aromaticity indices: RE/e, RE/a and REPA are given for a number of conjugated dianions in Table 5. To further illustrate the comparison, the plots RE/e vs. REPA and RE/a vs. REPA are given in Figs. 6 and 7, respectively. The points in both figures are

Fig. 5. A plot of RE (in eV ) against TRE (in /I) for polycyclic conjugated dianions for which both results are available. Satisfactory correlation is seen, except for dianions XXII and XLIII.

266 TABLE 5 Comparison between the RE/e and REa values obtained in this work and the REPA values of Bates et al. [ 35 ] for some polycyclic conjugated dianions Compound VII VIII IX X XI XII XIII XIV xv XVI XVII XVIII xx XXI XXII XXIII XXIV xxx xxx1 xxx11 xxx111 XXXIV xxxv XXXVI XXXVII XXXVIII XXXIX XL XL1 XL11 XL111

RE/e”sb (eV) 0.011 0.034 0.036 0.045 0.050 0.0518 0.0524 0.058 0.047 0.062 0.066 0.066 0.012 0.023 0.031 0.003 0.036 0.107 0.021 .0.009 0.074 0.091 0.052 0.010 0.0752 0.0754 0.017 0.061 0.023 0.064 0.079

RE/a+” (eW 0.014 0.039 0.041 0.050 0.055 0.0575 0.0582 0.064 0.052 0.068 0.072 0.073 0.015 0.027 0.036 0.003 0.040 0.133 0.025 -0.011 0.087 0.106 0.061 0.011 0.0859 0.0862 0.019 0.068 0.026 0.071 0.089

REPfid (B) 0.039 0.064 0.045 0.068 0.059 0.048 0.055 0.040 0.062 0.057 0.056 0.068 0.042 0.070 0.042 0.043 0.045 0.116 0.050 0.033 0.099 0.108 0.081 0.046 0.094 0.094 0.044 0.069 0.032 0.083 0.096

“This work. bRR/e = resonance energy per electron. %E/a = REPA = resonance energy per atom. dRef. 35.

rather scattered, but, in most cases, the smaller RE/e or RE/a values correspond to smaller REPA values and, conversely, larger RE/e or RE/a values correspond to larger REPA values. Therefore all three indices classify dianions in roughly the same way, although they differ in the prediction of the degrees of conjugation. For example, the triphenylene dianion (XIV) is predicted to be more aromatic by RE/e (0.058 eV) and by RE/a (0.064 eV) than by REPA (0.040 /3). The question concerning the degree of aromaticity of dianions ap-

257

0

1. 0

w 0

o

oal

XIV 0

00

QSa-

0Ql

Q

0

O”a

*

0

0

a3 4-

0 0

i 0

, a54

m

REPAl I 3

Fig. & A plotof RE/e (in eV) vs. REPA (in 8) for come polycyclic conjugated dianions for which both sets of results are available. Fig. 7. A 6,

plotof RE/a (in eV ) ve. REPA (in $3) for the came Bet of conjugated dianions as in Fig.

pears here to be only academic. However, in our recent test [ 10 ] of 25 different theoretical methods for computing resonance energies, all methods based on the HMO, such as the TRE approach or the HS approach, failed the test for innate consistency. The test for innate consistency was as follows, RI, R, and & circuits were calculated from the REs reported for benzene, naphthalene and anthracene by each method and if the test for innate consistency was successful then the minimum condition for the consistency was fulfilled, Le. R, > R, > R,. Thus, it appears that HMO-based approaches have some unresolved difficulty. This so-far ~nr~o~~@~ difficulty may be at the bottom of the very rough correlations between RE/e and REPA (Fig. 6 1, and /a and REPA (Fig. 7). Figures 6 and 7 are rather similar, giving no preference to either RE/e or RE/a. A plot of RE/e vs. RE/a for studied dianiona is linear (see Fig. 8). Thus, either index can be used as an aromaticity index for conjugated dianions. In view of the virtues of the HMO [ 91-93 f method, close comparisons between advanced and ahernative approaches with those of HMO seem desirable. Comparison with the approach using the concept of conjugated circuits may be particularly useful in giving some insight as to why the HMO method works so well in situations when one is not interested in single orbital contributions but total molecular quantities, such as RE. An illustration of such validity of the HMO approach is demonstra~d by a very good ~~~~e~a~ion between HMO molecular energies and log K, K being the number of Kekul4 va-

268

Fig. 8. A plot of RE/e (in eV ) vs. RE/a (in eV ) for all the studied polycyclic conjugated dianions.

- _m m_

ti(Rt+Q,t

2 &I

2R,+Q,

2fQ, +a3

-2(Rt+Q,l

Fig. 9. The circuit decomposition for the valence structures of butalene dianion.

lence structures, the correlation [94] which, once again, reminds us of the intri~ing in~rl~king between simple MO and simple VB schemes [ 95,96]* It seems, therefore, that, besides a rather apparent desire of quantum chemists to produce better and better numerical computations, useful information about molecular structure can be extracted by comparison of simple approaches with more elaborate ones [ 971. Hence, a more systematic search for areas of agreement and disagreement between different theoretiical models seems desirable and such work is contemplated [ 981. We conclude this section with a brief discussion of another bicyclic conjugated dianion: C, Hi- (see Fig. 9 ) . In view of the fact that a four-membered ring characterizes the structures for which some disagreement was observed between the HMO approach and the SCF MO approach, the above system is likely to clarify the issue further. According to the circuit decomposition, the RE of butalene dianion is given by

RE= (8& +4Q1 +9Q2)/11

(6)

which gives RE=O.166 eV, RE/e=0.021 eV and A=0.15 (15%). Thus, the butalene dianion is predicted to be slightly aromatic. The REPA value of butalene dianion ( -0.027 /3) is in the range of slightly anti-aromatic structures [ 351. The parent hydrocarbon butalene is predicted to be an anti-aromatic [RE= (2&+4Q,)/3; RE= - 0.462 eV, AA = 0.291and the experimental data support this prediction [991. The peripheral model would, of course, predict butalene to be aromatic because its periphery contains six electrons. The model would not take into consideration smaller cyclic fragments containing four n electrons which destablilize the structure [99]. Similarly, the model would predict the butalene dianion to be anti-aromatic because the periphery of the molecule now contains eight K electrons. Neither prediction is supported by other theoretical models of aromaticity or by the scarce experimental data. The butalene dianion is unknown. However, we expect this dianion to be accessible and not very dissimilar to other structures with A < 0.2, such as the benzocyclobutadiene dianion (A = 0.16). One advantage of our graph-theoretical approach is its rather simple and quick analysis which allows an advance screening of a collection of related compounds to see which of them may exhibit less usual structural traits, even some novel structural features. We have outlined one such examination of polycyclic cations elsewhere [ 77,100]. This present work suggests a systematic search for discrepancies between different theoretical models as a methodology for identifying more interesting polycyclic conjugated structures or their ions and, subsequently, once the compounds become available, for assisting in the location of critical structural components which may be responsible for the differing predictions of models which generally give similar results. CONCLUDING REMARKS

The graph-theoretical approach to aromaticity and conjugation outlined here on dianions of polycyclic conjugated hydrocarbons appears to have all the ingredients of a valid model: (i) it is conceptually simple; (ii) it incorporates some components of previous methods (such as the 4n + 2 count of n electrons, now confined to selected circuits and the use of a molecular perimeter when it qualifies); (iii) it allows numerical computations; (iv) it clarifies the aromaticity notion and confusion; and last, but most importantly, (v) its results agree with experiment. Significantly, the model can be used to deduce valid conclusions solely by the inspection of molecular valence structures, whether one is interested in a single structure or in a comparison of several structures. The approach offers a more powerful language for discussion of the relative differences among related structures. For example, several compounds in Fig. 4 have

270

the same periphery and a simple criterion, even if it would be valid, could not distinguish between such compounds. Comparison of isomeric structures and of structures having the same number of 7celectrons exhibits the power of the conjugated-circuits model. For example, zethrene and its non-Kekulean isomer (which has no classical Kekuletype valence formula as a neutral system) have the same perimeter. The dianions of the two structures XLVI and XLVII are particularly interesting. They cannot be differentiated between by the peripheral model and, in view of the 22 peripheral carbons, we have a dilemma: should we add two negative charges to the molecular periphery making it a 4n type or should the negative charge reside on the inner carbons?

@??= XLVI

@Ia= XLVII

RE (XLVI) = (958 RI + 434 R2 + 43 R3 + 108 Qz + 186 Q,)/484 = 1.887 eV A (XLVI) =0.917 or 91.7%

RE (XLVII) = (776 R, + 368 R2 + 48 R3 + 40 Q3)/386 = 1.867 eV A(XLVI1) =0.991 or 99.1% The full analysis shows that there are a large number of valence structures in both dianions and that they are almost aromatic, their A values being above 91%. This may superficially suggest that the charge is located on the inner carbons, as this would leave the 22 x electrons of the periphery as critical in determining the molecular characteristics. However, in fact, a small fraction of valence formulae have charge in the centre, and a very few have both charges simultaneously at the inner carbon atoms, hence the charge is predominantly distributed all over the molecule, including the perimeter. The difficulty is resolved if one recognizes that small conjugated rings and circuits make important contributions to the RE and molecular stability. Hence, even if there are 4n x electrons in the periphery, as is the case with the 16 A electrons of pyrene which was discussed in the opening section of this paper, this is not crucial for determining molecular characteristics. The crucial structural element is the distribution of the n electrons in the conjugated circuits, because RE is additive in terms of these qualified circuit contributions. NOTE ADDED IN PROOF

The TRE value of XL111 is 0.573p [ 341. The TRE value of - O.O33/?inTable 4 was taken by mistake from the paper by Ilic and Trinajstic’ [ 341 for the TRE value of XLIII.

271

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273 82 83 84 85 86 81 88 89 90 91 92 93 94 95 96

97 98

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