The Correspondence between the Resonance and Molecular Orbital Theories

The Correspondence between the Resonance and Molecular Orbital TheoriesAuthor(s): M. J. S. Dewar and H. C. Longuet-HigginsSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 214, No. 1119 (Oct. 9, 1952), pp. 482-493Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/99013 .

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The correspondence between the resonance and molecular orbital theories

BY M. J. S. DEWAR, Queen Mary College, London AND H. C. LONGUET-HIGGINS, University of Manchester

(Communicated by C. A. Coulson, F.R.S.-Received 20 September 1951- Revised 8 May 1952)

The resonance theory of organic chemistry is critically examined from a theoretical view- point. It is stressed that this theory is not rigorously founded on the quantum-mechanical valence bond method, but involves additional assumptions which cannot be defended. The practical success of the resonance theory must therefore be explained in some other way. It is here shown that a remarkable correspondence exists between the resonance theory and the molecular orbital method; and it is suggested that the resonance theory owes its success more to this correspondence than to the validity of its own premises.

1. INTRODUCTION

As is well known, it is impossible at present to apply wave-mechanical methods exactly to the solution of chemical problems. Considerable progress has, however, been made with the aid of two distinct approximate methods, the valence bond (VB) method, and the molecular orbital (MO) method. Within their own limitations both these methods have been rigorously applied to a number of systems, and with a few understandable exceptions they lead to the same general conclusions.

The formalism of the valence bond method bears an obvious relation to the con- cepts of the classical theory of organic chemistry; and this led chemists to try and base the classical theory upon an intuitive generalization of the valence bond method. This intuitive generalization, known as the resonance theory, requires for its practical application the use of three further assumptions beyond those already contained in the valence bond method. These assumptions are, in effect (Wheland I944):

First that the wave function of a molecule can be represented sufficiently well as a linear combination of wave functions for the unexcited canonical forms or classical structures. This implies that if there is only one classical structure, the wave function corresponding to this structure should represent the molecule adequately.

Secondly, that if there are two or more classical structures, the molecule will have an intermediate steric configuration, and an intermediate reactivity- tempered by an additional stability or resonance energy relative to the most stable classical structure.

Thirdly, the stability of a mesomeric molecule, i.e. a molecule possessing several possible classical structures, is greater, the greater the number of such structures.

These three assumptions have very little basis in theory, as various authors have pointed out. Two criticisms are particularly serious. First, there is no good justification for neglecting the excited canonical forms; although the contribution of a particular unexcited canonical form may be greater than that of any one excited

[ 482 ]



Correspondence between resonance and molecular orbital theories 483

canonical form, this may well be outweighed by the greater number of excited canonical forms. Indeed, calculation confirms this in polycyclic aromatic hydrocarbons, the unexcited canonical forms (classical structures) making virtually no contribution to the ground state of naphthacene (Pullman & Pullman 1946).

Secondly, even if the classical structures really had a paramount importance, there would be no justification for relating the stability of the molecule to their number; the stability depends not only on the number of canonical forms, but also on the interactions between them, and these are by no means constant. Similar objections can be raised to the use of the resonance theory for estimating such quantities as charge distributions and bond orders in molecules.

These criticisms, it must be stressed, do not- apply to the valence bond method itself, but only to the three distinctive assumptions which characterize the resonance theory. Yet in spite of these objections the resonance theory works well, and it has become increasingly difficult to understand why this should be so. In this paper the problem is examined from a different point of view, and attention is drawn to the existence of a remarkable 'correspondence principle' between the resonance theory and the MO method. It seems likely that the success of the resonance theory is largely due to this unexpected correspondence.

2. RECENT WORK

By the application of perturbation theory to the MO equations, a general theory of organic-chemical behaviour has recently been developed (Longuet-Higgins 1950;

Dewar I952). This, the MO counterpart of resonance theory, not only covers the same ground, but provides in many cases semi-quantitative information which the resonance theory could not provide. It is, moreover, free from the subjective element which is so difficult to avoid in any application of the resonance theory. The following results have been established by this MO treatment:

(1) If only one classical structure can be written for a molecule, the molecular orbital bond orders will show the expected alternation between 'single' and 'double'.

(2) If it has more than one classical structure, the molecule will assume an intermediate configuration, and will usually be more stable than an isomer with only one classical structure.

(3) The distribution of charge in an unsaturated carbonium ion or carbanion such as triphenyl methyl corresponds qualitatively to that given by resonance theory, and the same is true of the distribution of free radical character in unsaturated hydrocarbon radicals.

(4) The effects of substituents on the charge distribution and chemical behaviour of alternant compounds (see ? 3) correspond qualitatively to the predictions of resonance theory; and the same is true of the effects of replacing carbon atoms by hetero-atoms.

(5) Compounds with no ordinary classical structures should be abnormally unstable.

The above results justify the conclusions of the resonance theory in so far as they show that these conclusions can be reached by an alternative and sounder theo-



484 M. J. S. Dewar and H. C. Longuet-Higgins

retical approach. It remnains to be explained, however, why the resonance theory leads to correct conclusions, in the face of apparently devastating objections to its premises. The answer seems to lie in an unexpected and almost fortuitous correspondence between the resonance and MO theories.

3. RELATION BETWEEN THE THEORIES

The following discussion will be confined to alternant systems (Coulson & Rushbrooke I940; Longuet-Higgins I950), that is, systems in which alternate conjugated atoms can be labelled with stars in such a way that no two starred atoms are directly linked, and no two unstarred atoms. Most conjugated molecules are of this type. It can be shown that in an odd alternant hydrocarbon (AH) such as the benzyl radical, the topmost occupied MO, which contains the odd electron, has zero binding energy, and is confined to alternate atoms only. For definiteness we shall suppose that the starring has been performed in such a way that the starred atoms outnumber the unstarred atoms by one; with this convention it can be shown that the non-bonding MO is confined to the starred atoms, and takes the form

Ea,O, (r starred), (1) r

where Or denotes the appropriate atomic orbital on atom r. Consider any classical structure of such an odd AH. All the conjugated atoms are

linked in pairs by double bonds except for one (starred) atom, say atom r. Let the number of classical structures with the odd atom in this position be nr. We are now going to show that there is a close connexion between the coefficients ar and the numbers nr.

For this it is necessary first to divide the classical structures into two classes, which, for reasons which will become clear later, we call 'positive' and 'negative'. We want to arrange that two structures are of opposite 'sign' if one can be obtained from the other by shifting just one double bond. The following specification ensures this result. Number the unstarred atoms from 1 to h, and the starred atoms from h + 1 to 2h + 1, in any order. Then in any classical structure all the starred atoms are paired with unstarred atoms, except atom r, which we shall regard as being paired with a fictitious unstarred atom 0. The structure in question is then defined to be positive or negative according as atoms 0 to h are paired with an even or an odd permutation of atoms h + 1 to 2h + 1.

Let us introduce the letter ur to denote the number of positive structures with the odd atom in position r, minus the number of negative structures with the odd atom in the same position. We may call ur the 'algebraic' number of classical structures in which atom r is the 'odd' starred atom; and we shall now show that the numbers Uqr satisfy the secular equations for the non-bonding MO. Let r, t and u be the starred neighbours of any unstarred atom s. Then in the immediate neighbourhood of atom s any structure in which atom r, t or ui is odd will have one of the six forms shown in figure 1. To every structure of type (a) there corresponds one of type (a'), in which one double bond has been displaced; these two structures are necessarily of opposite sign. The same applies to (b) and (b') and to (c) and (c'), if such structures




exist. It follows that the positive and negative structures of all these six types are equal in number, i.e. that Ur+Ut+Uu 0. (2)

But the corresponding secular equation for the non-bonding MO coefficients is just

ar +at+au?O, (3)

r t

(a) (b) (c)

(a') (b') (c')

FIGURE 1

there being one equation of type (2) and one of type (3) for every unstarred atom in the system. It follows that the normalized coefficients in the non-bonding MO are given by ar U 2r(r)- (4)

r

4. BENZENOID SYSTEMS

The numbers ur in equation (4) have been somewhat abstrusely defined; but there is an important case in which equation (4) may be simply interpreted. If all the structures with the odd atom in position r are of the same sign, then

r n =r (5)

and so ar=?nr(Enr2)Ai, (6) r

the sign depending on whether these structures are all positive or all negative. This situation occurs, for example, in the benzyl radical, where the values of nr are a, are as shown in figure 2. We therefore enquire: under what conditions will all the structures in which atom r is odd be of the same sign?

o 1 - 1/N/7

02 1/A/7 0 /V/7

o 1 o -11V7

FIGURE 2

In answering this question we may ignore atom r completely, since its presence makes no difference to the disposition of double bonds in the rest of the molecule. It turns out that provided the residual molecule is built up entirely from chains and




(4n + 2)-membered rings joined or fused together, then all the structures will indeed be of the same sign. The proof is as follows:

Consider any particular structure (A) in which the unstarred atoms have been labelled in any order from 1 to h. Denote the starred atoms which are paired with 1, 2, ... h by h + 1, h + 2, .., 2h, and the odd starred atom by 2h + 1. Then in any other structure (B) with the same atom unpaired, atoms 1, 2, ..., h will be paired with some permutation P of h + 1, ..., 2h. Now any permutation whatever can be analyzed into a product of cyclic interchanges; for instance, the permutation 12345 -? 35412 is a combination of the two cyclic permutations (134) and (25). In the present context a cyclic permutation of this kind corresponds to a cyclic interchange of unstarred partners between a set of starred atoms. Now a cyclic interchange of this kind is only possible if the starred atoms and their unstarred partners form a closed ring (R) of alternate single and double bonds; if R contains an odd number of starred atoms the cyclic permutation is even, and vice versa. Let us ignore for a moment the atoms outside the ring R. Then R forms the perimeter of an even AH. Now it is quite easily proved by induction that if this AH is built up solely of hexagons or other (4n + 2)-membered rings then its perimeter must contain 4m + 2 atoms, of which, naturally, 2m + 1 will be starred. Therefore, provided there are no 4n-membered rings present, the cyclic permutation R and the more general permutation P will both be even, so that the arbitrary structures A and B will have the same sign, and equations (5) and (6) will hold. This is what we set out to prove.

FIGURE 3

In applying this result there is one pitfall which should be mentioned. The AH radical in figure 3 is composed of three hexagons; but the two structures in which the central atom is unpaired are not of the same sign. This is because removal of the central atom leaves an open 12-membered ring, and 12 is a multiple of 4. This situation always arises for internal atoms in benzenoid radicals; but provided atom r is on the perimeter equation (5) may be used without error.

5. INTERACTION BETWEEN CONJUGATED RADICALS

To sum up, we have shown (i) that the non-bonding MO in an AH radical has coefficients given by ar = Ur( ur2)-4, (7)

r

and (ii) that if the molecule left on removal of atom r is benzenoid (i.e. contains no 4n-membered rings), then nr ur I so that

ar = + const. nr.

The above results are exact. However, in order to apply them to specific problems it is necessary to appeal to an approximate equation connecting the coefficients ar




with energy quantities. The connexion arises in the following way. Suppose we have two hydrocarbon fragments A and B, and join them together to form a conjugated molecule AB. Then there will be a certain 'conjugation energy' between A and B, defined as the difference in unsaturation energy between the fragments and the united molecule. It can be shown that if every unsaturated bond in A and B has its resonance integral equal to fi, and if x,8 is the resonance integral of the bond(s) joining A to B, then as a rule (Coulson & Longuet-Higgins I948) the conjugation energy is of the second order in x, and takes the form

AE=EAB-EA-EB = fl.O(X2). (8)

There is, however, a special case in which AE is of the first order in x, namely, when A and B are both radicals. This may be seen as follows (Dewar I952). Let the MO of the unpaired electron in A be

WA4 = Ear,r, CA = ', (9) r

and that of the unpaired electron in B

VfBY= E bsOvs CB = ? (10) S

Now suppose that atom p of A is joined to atom q of B by a weak bond. Then VfA and ifB, being degenerate, will split into the two combinations

f = V12 (VA+B)} (11)

Vf- = 1 2 ( RIA 3 VB ), with energies

6+= 2 (6Ae + 6B) )pq ap bq= x/ap bqA1

6-= 2 (CA + ?B)-/Pq ap bq = -x/ap bq. (

The most stable state of A ... B will be that in which the originally unpaired electrons have both gone into Vf or Vf_, whichever is the more stable. Therefore the contribution of these two electrons to the conjugation energy is just 2xf I a bq The total conjugation energy is therefore

AE = 2xl I apbq +?/?.O(x2), (13)

and the mobile order of the bond pq is

_ aEAB I 1 AE Ppq 2 I/a_ = 2pl ax = Japbj+0(x). (14)

(If A and B are joined by several bonds pq, p'q', etc., the generalization of (13) is

AE= 2x/?Iapbq+ap,bq,+...?/ .O(x2), (15)

and in the particular case that B consists of one atom only,

AE=2x/ I aP + a., +... I +I ?. O(x2), (16)

where p, p', ... are the atoms of A which are bonded to it.)




The above equations show that when x is small the contribution of the odd electrons in A and B to the conjugation energy outweighs the contribution of the other electrons. Actually, the first-order term, when it occurs, dominates the situation even when x = 1. We shall therefore make the approximation of dropping the second order terms entirely, and write

ZXE =2,C apbq+ ... (17)

and Ppq ap bq1. (18)

6. APPLICATIONS

(a) Aromatic substitutionf in an AH

According to present ideas (Wheland I942) the ease of substitution in an AH may be related to the difference in total 7r-electron energy betweell the AH and a transition state, in which the reagent has attached itself to a carbon atom which consequently becomes tetrahedral and is removed from conjugation. This difference in rr-electron energy is the difference between the energy of the parent even AH and that of the odd AH derived from it by removal of one atom. The resonance theory states that this difference will be less, and reaction will therefore occur more readily, the more classical structures that can be written for the transition state. Now the number of classical structures is clearly the same for substitution by all types of reagent, ionic or radical; the orientation of substitution in an AH should therefore be independent of the reagent. The same conclusion follows in MO theory from the fact that all three types of transition state have identical sT-electron binding energies, since they differ only in the numbers of electrons occupying the non-bonding MO; the contribution of the ir-electrons to the activation energy is therefore the same for all types of reagent. This conclusion is supported by the chemical evidence.

r r r

.C-H C SC=-C ZC-H ~~ ~~/ \X "C( - CH2 - 2

(I) (II) (III) FIGURE 4

Consider now the transition state (II) from an even AH(I) in which atom s is linked to atoms r and t. The in-energy difference between (I) and (JI) is, with a change of sign, equal to the conjugation energy between atom s and the rest of the molecule. Therefore to a first approximation,

E(I)-E(I)- AE =2 ar+a (19)

where ar and at are the non-bonding MO coefficients for (II) and 18 is the carbon- carbon resonance integral. Denote by Uk the algebraic number of structures of (II) in which atom k is unpaired. Then

AE 2fl 1 ur + u I (Ju.r)-'. (20) r




Now consider the odd AH (III) obtained by adding an extra CH2 group at position s (for clarity only the conjugated atoms are shown). To every structure of (I) there is just one structure of (III) with the double bonds disposed in the same way and atom u unpaired. If we suppose (I) to be benzenoid, then all its structures are of the same sign; and the same applies to the structures of (III) in which atom u is unpaired. Further, structures of (III) in which the odd atom is r or t differ from those with u odd by the displacement of just one double bond, and are therefore of opposite sign to the latter. Denoting the algebraic numbers of these three kinds of structure by vr, vt and v., and using the fact that Vr + vt + v. = 0, we obtain

|Vr + I Vt I = IVr+Vt I 1V. (21) Now in structures of (III) with r or t odd, s must be doubly linked to u, and the presence of these two atoms is therefore irrelevant to the disposition of double bonds in the rest of the molecule. It follows that

1|Vr I + I Vt I = I Ur I + IUJt I = I Ur+Uji 1 (22)

For a similar reason I V. I is just the total number of classical structures of (I). Combining equations (20), (21) and (22) we thus obtain

AE = -2flIvuI ( ur2)-i r

= const. (Eu2)-i. (23) r

Therefore, according to this approximate formula of MO theory, substitution should occur more readily, the larger E ur, since an increase in this quantity lowers AE.

r

But in resonance theory substitution is supposed to occur more readily the greater the number of classical structures for the transition state, i.e. the greater Y nr.

r

These two statements, though different, amount to roughly the same thing if

nr = ur I for all positions in (II); for then

E U2 =En n2 (24) r r

and in general E nr will be greater, the greater E n2. Provided (I) is benzenoid, r r

nr will be equal to ur for all except internal atoms, which are seldom numerous; so E r2can differ from E n2 by at most one or two terms in the sum, and this only in r r

cases where the total number of terms is large. Therefore there should be, for benzenoid hydrocarbons at any rate, a correlation between the predictions of the two theories.

(b) The ground states of AH's

It is always tacitly assumed in resonance theory that any conjugated hydrocarbon should be capable of preparation provided (i) it is not too highly strained or sterically hindered and (ii) it has at least one classical structure. This assumption is based on much experience of benzenoid hydrocarbons, and has provided the impetus for repeated, but hitherto unsuccessful, attempts to prepare cyclo- butadiene.




Longuet-Higgins (I950) recently showed that an even AH possessing only excited canonical structures should, according to MO theory, behave as a biradical. We shall now show that for benzenoid hydrocarbons the possession of one or more unexcited structures is a sufficient, as well as a necessary, condition for the molecule to have a normal ground state. Our criterion for a normal ground state is the absence of non-bonding MO's, because whenever these occur one IT-electron will tend to go into each, and the molecule will behave as a biradical.

Expansion of the secular determinant for an even AH yields a polynomial in 62,

where c measures the binding energy of a MO. The presence or absence of non- bonding MO's depends entirely on whether the constant term in this polynomial vanishes or not; if it vanishes then 62 is a factor of the determinant, so that the secular equations have two zero roots. Now the constant term is just the square of a smaller determinant, each row of which corresponds to a starred atom and each column to an unstarred atom, the element in any row and column being ,8 or 0 according as the corresponding atoms are adjacent or not. Each term in the expansion of this determinant is a product of the resonance integrals for the double bonds in some unexcited structure, the sign of the structure determining the sign of the product. The value of the determinant is therefore A8hu, where u is the algebraic number of unexcited structures of the AH; and the constant term in the secular polynomial is /32hU2

Now for a benzenoid AH all the unexcited structures are the same sign, so U2-n2, where n is the arithmetic number of them. Therefore if a benzenoid AH has one or more unexcited structures the constant term is positive, so there are no non-bonding MO's. In non-benzenoid AH's, on the other hand, u may vanish without n doing so. For instance, in cyclo-butadiene there are two unexcited structures; but they are of opposite sign, so that ua vanishes and there are two non-bonding MO's. In spite of having two classical structures this hydrocarbon would probably behave as a biradical; and qualitative arguments from resonance theory as to the stability of such non-benzenoid hydrocarbons should be treated with reserve.

In applying these results it should be borne in mind that our criterion for a normal ground state is not quite foolproof. If a conjugated molecule has its topmost occupied and lowest unoccupied MO very close together, repulsion of the two electrons in the former may cause one of them to be excited to the latter, in spite of the promotion energy involved. The occurrence of two coincident non-bonding MO is merely a special case of this. We might expect, therefore an even AH to exhibit radical character when the constant term ,/2hu2, though not actually zero, is small for a molecule of that size. Thus a large even AH with only one or perhaps two unexcited structures will be quite likely to behave as a biradical, particularly at high temperatures. A corollary of this is that a large even AH with very few unexcited structures will tend to have excited electronic states close to the ground state. This generalization agrees with the empirical rule that AH's whose resonance structures are quinonoid tend to be strongly coloured; for if a system is quinonoid it tends to have rather few classical structures, because of the impossibility of per- muting the bonds in a quinonoid ring.




It also follows that the chemical reactivity of an even AH should be greater, the fewer the unexcited structures that can be written for it; even apart from the possibility of thermal excitation to biradicals. For Dewar (I952) has shown that the energy difference AE in (19) should be approximately proportional to the minimum excitation energy. This generalization is also supported by the available evidence. Thus the polyacenes absorb at much lower frequencies and show much greater chemical reactivity than do isomeric 'non-linear' hydrocarbons; and more unexcited structures can invariably be written for the latter.

In short, we have given reasons why benzenoid AH's with several unexcited structures should have normal ground states, why large AH's with very few unexcited structures should have low-lying excited states (such AH's often being quinonoid in structure), and why even AH's with no unexcited structures should behave as biradicals. All these results have hitherto had the status of purely empirical rules in resonance theory.

(c) Charge distribution in odd AH ions

In resonance theory the charge at an atom in an odd AH ion is supposed to be greater, the greater the number of unexcited structures with the odd (charged) atom located there; in other words, the net charge on atom r is supposed to be proportional to nr. MO theory, on the other hand, gives the value of this charge* as a2. But in a benzenoid AH, by equation (6), ar is proportional to n2, which varies from atom to atom in the same direction as nr. Hence the resonance theory does account in a qualitatively correct manner for the distribution of net charge in benzenoid AH ions. This argument does not apply, however, to non-benzenoid systems.

(d) Bond orders

In the crude application of resonance theory, the mobile order of a bond is equated to the fraction of unexcited structures in which it is written as double; and this simple measure of bond order, combined with a semi-empirical order- length curve, predicts bond lengths surprisingly well for many aromatic systems. Our 'correspondence principle' suggests the following interpretation of this:

Consider a benzenoid AH R, and the odd AH Br obtained from it by removing one atom r. Let the non-bonding MO coefficients of Br at the atoms s, t, u attached to r be a8, at, a.. Then to a first approximation the orders of bonds rs, rt and ru in R are, by (18), Prs a.1, Prt =atl, Pru Iau l (25)

Therefore Prs :Prt :Pru | 1'1 Urt 1 Uru 1 (26)

where urs is the algebraic number of unexcited structures of the AH Rrs obtained by eliminating both r and s from R. (If Brs is a pair of radicals or a biradical, then of course urs = 0.) Now since R is, by hypothesis, composed entirely of chains and

* An AH radical, containing one electron in a non-bonding MO, has zero net charge at every position. Removal of this electron gives a positive AH ion, and addition of another electron to the same MO gives a negative AH ion. Hence the distribution of charge in these ions, like the distribution of the odd electron in the radical, is determined solely by the non-bonding MO coefficients ar.



49$2 M. J. S. Dewar and H. C. Longuet-Higgins

hexagons, 1rs will also be composed of chains and hexagons (and possibly one 14-membered ring if rs is a completely internal bond). Therefore

I U rs I - nrsI u |rtI , U ru I = nru'

where nrs is the arithmetic number of unexcited structures of 1? in which bond rs is written as double. Hence the mobile orders of the three bonds are approximately in the ratios of the numbers of classical structures in which they appear double.

Two points require comment. First, equation (25) is only valid if a8, at and a. are of the same sign. This condition is satisfied if R is benzenoid, but not necessarily otherwise. Secondly, what we have done is to show that the molecular orbital bond orders are approximately given by the procedure which is commonly used for estimating 'double bond character' in resonance theory. Not only does this set the idea of double-bond character on a much sounder basis, but it also explains the rather puzzling fact that MO bond orders usually form a pattern reminiscent of the classical bond pattern, or a superposition of such patterns if there is more than one.

(e) Relative stabilities of isomeric AH's

A classical argument of resonance theory was that phenanthrene would be expected to show greater stability than anthracene because five classical structures can be written for the former, but only four for the latter. This kind of argument had no theoretical basis, but again its success can be understood in terms of the present treatment.

Consider an even AH formed by the union of two odd AH's A and B, atoms p and q in A being joined respectively to atoms r and s in B. Let the non-bonding MO coefficients of A and B be denoted by ai and bj respectitely. Then the total IT-energy difference between AB and the separated pair A +B is given approximately (cf. equation (15)) by

AE 218Japbr+aqbsj. (27)

Now let mp, mq be the numbers of unexcited structures of A with the odd atom at positions p, q respectively, and let nr, ns be similarly defined for B. Then provided AB is benzenoid, (27) may be written in the form

A E = 2?CAC CB(mp nr + mqns), (28)

where CA and CB are constants characteristic of the fragments A and B. But mp nr is just the number of unexcited structures of AB in which bond pr is double and bond qs single, and conversely for mq n8. And evidently in any unexcited structure for AB one or other of these bonds must be double, but not both. So the quantity in brackets in (28) is just the total number of unexcited structures for the united molecule.

This result shows that if the fragments A and B can be joined together in more than one way to form a benzenoid AH, then the resulting molecule will be more stable the greater its number of unexcited structures. Figure 5 shows some ex- amples of the ways in which two benzyl radicals can be joined together to form benzenoid AH's; the number of unexcited structures of each combination provides a valid criterion of their relative stabilities.




4 structures 1 structure

2 structures

4 structures 5 structures

FIGuRE 5

The above demonstration holds rigorously only for pairs of AH's which can be interconverted by fission and formation of not more than two bonds; but it is likely to hold approximately in other cases also.

7. CONCLUSION

By clarifying the relation between the resonance theory and the quantum- mechanical molecular orbital method, this analysis seems to provide an explanation for the surprising success of the resonance theory in systems composed of chains and odd-membered rings. This is satisfactory, because its success cannot be explained by arguments based on the valence bond method; and still more because the possibility now lies open of translating the ideas of resonance theory into a more rigorous formalism based on the molecular orbital method.

REFERENCES

Coulson, C. A. & Longuet-Higgins, H. C. I948 Proc. Roy. Soc. A, 195, 188. Coulson, C. A. & Rushbrooke, G. S. I940 Proc. Camb. Phil. Soc. 36, 1931. Dewar, M. J. S. I952 J. Amer. Chem. Soc. 74, 3341, 3345, 3350, 3353,

3355, 3357; J. Chem. Soc. (in the Press). Longuet-Higgins, H. C. I950 J. Chem. Phys. 18, 265, 275, 283. Pullman, A. & Pullman, B. I946 Experientia, 2A, 364. Wheland, G. W. I942 J. Amer. Chem. Soc. 64, 900. Wheland, G. W. I944 The theory of resonance. New York: John Wiley.

Vol. 2I4. A. 32



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The Correspondence between the Resonance and Molecular Orbital Theories