Resonance Theory and the
William C. Herndon Un~versityo f Texas at El Paso El Paso, Texos 79968
Enumeration of Kekule Structures
"Use resonance theory to predict the relative stabilities of the isomers, anthracene and phenanthrene." This type of examination question is a familiar one to students engaged in the study of r-molecular systems. Nearly all elementary and advanced textbooks of organic chemistry give interpretations of stabilities of molecules in terms of the number of reasonable valence-bond structures that one can draw. Energies of unstable intermediates are also ordered on the basis that increased stabilities and associated resonance energies are correlated with increasing number of structures. The molecular species under discussion is said to he a resonance hybrid of the contributing structures, and the use of resonance theory leads to valid predictions of relative stabilities and reactivities. The only serious defect of the theory is a prediction of large resonance energies for monocyclic r-systems containing 4n(n = integer) electrons. The formulation of resonance theory as it is practiced today is explicated in the well-known books by Pauling (I) and Wheland (2). Study of these texts shows that resonance theory is derived from quantum-mechanical valence-bond studies mainly carried out in the 1930's (3-6), and therefore the theory has a mathematical foundation. However, the approximations required to reduce valencebond theory to resonance theory are so drastic that many theoreticians are loathe to ascribe validity to the less rigorous method (7-9). Most criticism is focused on the small number of structures that are included in resonance hybrids. In the case of the problem that prefaced this article there are four possible Kekule structures1 for anthracene and five structures for phenanthrene, shown in I. ~
~
w
structures. Longuet-Higgins and Dewar pointed out over 20 years ago several points of mathematical similitude (11, 12), and stated that the success of resonance theory was due to these "unexpected" results, where the word "unex~ected"was used in the sense of unforeseen. Thev certainly realized that mathematical identities between the theories must exist, although . they did not pursue that particular matter. The equivalencies between resonance and MO theory will he described in the next section. and then we will discuss several different methods for obtaining the structure count of r-molecular svstems. Ho~efullv.the results described will convince the reader that counting resonance structures is a sound theoretical approach. However, disregarding the chemical value of enumerating resonance structures, the problem has intrinsic mathematical interest. Several approaches have been attempted, some involving graph theory and comhinatorial mathematics. The subject has a long history, and is a good example of a problem in organic chemistry that can he solved by purely mathematical techniques. Isomorphism of Resonance and MO Theories Valence-bond theory and MO theory are both valid approximations to the quantum-mechanical treatment of a molecule. I t has often been pointed out that if extended calculations are made, including configuration interaction in the MO treatment and ionic structures in the valencebond approach (2, 13), the two methods yield identical results. This is not the similarity that will he delineated in this section. Rather, we will outline those essentially mathematical aspects of the theories that agree a t the simplest levels of application. This means that we will be comparing the results of Hiickel molecular orbital (HMO) calculations with resonance theory limited to Kekule structures. Closed Shells and Kekule Structures The concept of closed electronic shells is just as useful for organic r-systems as it is for inorganic ions. HMO theory (14) gives a set of energy levels called bonding (more stable than an isolated 2p-orbital), nonhonding, and antibondine levels. If each bondine level is douhlv occupied by electrons, with none of the other levels containing electrons, the molecular specieshas a closed shell. Benzene r-system is of this type. Three bonding and three conjugate antibonding levels are given by an HMO calculation, the three bonding levels being filled by the six r electrons. In general, closed shells are always calculated for altemant a-systems, the exceptions being monocyclic rings of 4n orbitals, and species for which one cannot draw a Kekule structure. An unsaturated hydrocarbon must possess Kekule structures to exist (15), and molecules of the type shown in I1 have proved impossible to prepare. The principle difference between these molecules and zethrene, shown in
-
I
These structures are taken to wholly represent the molecules, and the larger number for phenanthrene is considered to denote its greater stability. Yet these structures are iust a small fraction of the ~ossihlenon-ionic oaired electron structures. In both cases-there are 429 structures, and in results from rigorous calculations the contribution of Kekule structures to resonance energy is only about 8% (10).This fact raises feelings of dubiousness concerning resonance theory. I t also discourages one from carrying out valence-bond calculations on large molecules. There are circumstances that indicate that the use of Kekule structures alone is a more valid procedure than previously thought. First is the empirical fact that resonance theory works. The everyday utility of the method cannot be denied. Second, there are some precise mathematical congruities between some of the results of molecular orbital (MO) calculations and the number of Kekule 10
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III 'The words "Kekule structure" will refer to any valence-bond structure for unsaturated compounds in which single and double bonds alternate.
; is that no Kekule structures are possible for the former. Manv interesting examules are given in Clar's treatise on polycyrlic hyrlrocarhons~(lfi). .. Longuet-Higgins (11) was ahle to prove that the number of nonhondine enerm levels calculated hv HMO theorv fur alternant hy&ocarg&s is not less than N - 2T,where N is the number of D-orbitals. and T is the number of double bonds in a resonance structure. Further, for an altemant system each nonbonding level is occupied by a single elec&on, so that a ground state would have a multipl& structure. Each molecule in I1 would therefore have two unpaired electrons, and a triplet ground state. This is also exactly what resonance theory predicts since every principal resonance structure has two unpaired electrons. The proof that Longuet-Higgins adduced held only for benzenoid and open chain systems, and was graph-theoretical in nature. There are now many ways of determining whether or not an altemant molecule should be presumed to have a triplet or higher multiplet ground state, some of which will be described in the next section. The interesting methods from the standpoint of the present discussion are graph-theoretic methods for counting Kekule structures. For an even open chain or benzenoid rsystem, a zero structure count (SC) requires that at least two nonbonding levels be each singly occupied by electrons. The quickest way to determine if nonbonding orbitals are present in a benzenoid structure is to delete chains and rings of known stable structures from the molecule as far as possible. The number of remaining isolated atoms gives the number of nonbonding orbitals which is equal to the number of unpaired electrons. A structure should not be excised from a central part of the molecule so as to leave two odd fragments. The process is very simple so no examples will be given. If practice is desired, the reader may confirm that the structures in IV
n
N
have two and four nonbonding levels respectively. The recently defined corrected structure count (CSC) concept (17), which is only slightly more difficult to use, should be used for nonalternant and nonbenzenoid altemant systems. Structures and the Coefficients of NBMO's In any odd or even, .ahernant or nonaltemant hydrocarbon moiety, the coefficients of nonbonding (NB) MO's can be obtained without solving the secula; determinant (11). The sum of the coefficients must be zero around any orbital position (zero-sum rule). Extensive summaries of applications to several examples have appeared in the recent literature (18, 19), that outline a nonarbitrary reliable procedure. In a chain of orbitals (one orbital or longer) a zero coefficient is required a t penultimate positions, e.g., position 2 in V, positions 2 and 6 in VI.
-. '
-1-'
-
.I,
-*
a
VI
x-"fi
y-l/,&,
Other zero coefficients are assigned in accordance with the zero-sum rule. One remaining coefficient is given an arbitrary value of a. Using the zero-sum rule, other coefficients are given consistent values in units of a as far as possible. Further assignments of arbitrary coefficients ( b , c, etc.) may be necessary. If n independent parameters are required there are n independent NBMO's, whereas if only the trivial solution a = 0 exists there is no NBMO. The coefficients should be normalized and, in the case of the two degenerate levels for VI, orthogonalized. For completely cyclical systems an arbitrary assignment is made at any position, and solutions consistent with the zerosum rule signify a NBMO's. The odd electron density at each position in an odd alternant radical, or the charge for positive and negative ions isequal to the square of the coefficient of the NBMO at that position. The charge or electron density from resonance theory is greater the number of structures with the charge or radical electron located a t the particular point. Dewar and Longuet-Higgins (12) demonstrated, again with a graph-theoretic technique, that the absolute value of each MO coefficient exactly gives the number of structures with charge located at that position. Therefore charges are predicted by MO theory at the same points as by resonance theory. However, the absolute magnitudes of charge or electron density would differ, as demonstrated in VII for the benzyl ion, for which five principal resonance structures can be drawn. 8,:
1,s
MO Theory
Remnann Theory
W
The congruity between MO and resonance theories is not as satisfactory as one might wish. The exact mathematical equality is found between the number of resonance structures and the absolute values of unnormalized MO coefficients. The incongruity could be eliminated by assigning appropriate weights to the structures (201,but this would remove some of the simplicity of resonance theory. The exact relationship does allow one to enumerate SC for an odd alternant r-system quickly since all that one must do is obtain the sum of the absolute values of easily obtained NBMO coefficients. Therefore, a stu.dent can answer questions regarding the relative stabilities of ions and radicals quickly and accurately using regonance theory without drawing more than a single structure for each. Bond Orders Pauling bond orders (21) are defined as the number of structures in which a bond is double divided by the total .number of structures. In MO theory the concept of a bond order was introduced by Coulson (22) and is defined in esn. (l),
where the sum is over j occupied levels with n, representing the number of electrons in level j, and cjrcjs are MO coefficients for bonded atoms. Bond order as given by either theory is a convenient way of inferring the electron density in the region of a bond. When bond order is large the bond should be relatively short. Close parallels of bond orders and experimental bond lengths have been observed. For those compounds with accurately known bond Volume 51, Number
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lengths, both theories predict correct hond lengths to about k0.007 A (23, 24). There isalso an empirical linear relationship between Pauling and Coulson hond order (24). ,which of course accounts for the fact that hoth thedries are equally good in this respect. However, one wonders if a more definite connection between the theories might exist. An exact equality is described in some work on bond orders by Ham (25). He defines MO bond order so that the highest-filled MO levels contribute more to the bond order than the lower levels. The partial hond order of each level is weighted by the inverse of its energy as shown in eqn. ( 2 ) . P,. = ~njc;,c,, / E , (2) Roughly this hond order is proportional to the usual definition of MO hond order and is unexceptional in that regard. The main point is that the MO bond order defined in eqn. (2) is exactly equal to the Pauling hond order which shows that MO theory and resonance theory are in exact correspondence for at least one type of property. The hond orders defined in eqn. (2) can he written in terms of a matrix (and its minors) that is derived from the adjacency matrix of the graph of the molecule, examples to he given later. The point of interest is that the form of this matrix is a unique expression of the bonding and the branching properties of the r-system, and its value is invariant to the order of numbering orbital positions. Furthermore, the number of resonance structures for alternant systems is also given by a function of the adjacency matrix, the same function that appears in the denominator term of the matrix formulation for the bond order. So hoth the MO formulation of hond order, eqn. (21, and the definition of Pauling bond order can be expressed in terms of the same matrix. Platt sueeests that this conpruitv - between HMO theow and resonance theory has important chemical significance (26). We can have confidence that a theorv based on prim cipal resonance structures is likely to he at least as accurate as a Huckel-type MO calculation. The fact of this identity has much greater practical significance than the known but unobtainable identities between ab initio MO and valence bond theories.
all r'esonance structures for large molecules. Now a method has been published that allows one to count resonance structures and determine bond orders without writing the maioritv of the structures (17). Another noint.. havine to do k i t h the theoretical just;fication has already been k s cussed. The imposslbdit~ . of .iusfifvine - resonance theow on the basis of traditional valence-bondiheory is perhapsnot so important as believed formerly. A serious defect in Bartell's treatment is that it does seem to exaggerate resonance energies of nonhenzenoid structures. For example, azulene is calculated to have a greater resonance energy than naphthalene whereas all other theories and the thermochemical. evidence indicate that azulene has less than half the resonance energy of naphthalene (2, 29). SCF-MO calculations by Dewar and de Llano in which adjustments for compression and strain energy were made give azulene only 13% of the naphthalene value (30) in line with other recent MO calculations (31). No other nonalternant molecules have been treated by Bartell's model, so further discussion is not warranted. The work mentioned above by Dewar and de Llano (30) gives results for resonance energies that probably represent the best estimates of these quantities available. Their SCF-MO method is very successful in predicting heats of formation for several kinds of compounds, and their calculated resonance energies ought to be correspondingly accurate. Experimental estimates of resonance energies are difficult to define unambiguously. Consequently, in the most recent quantitative work on resonance theory (32), the standards chosen for comparison were the Dewar-de Llano values. In this work, only Kekule structures were included in resonance hybrids, and only two resonance integrals of the types shown in VIII and IX
.
A
-
Resonance Energies, a Quantitative Test of Resonance. Theory Quantitative applications of simple resonance theory are verv few in number. The theow is used extensivelv.. but primarily in a qualitative pedogogical sense where no mathematics is required. Two exceptional articles were written by Bartell (27, 28) in which he related a energies to Pauline series, deducina- bond orders by a simple - power . that resonance energies for a-electronic structures would he given by eqn. (3). In this equation
.
RE
=
(
(4/3)8 N
- XP,,'>
(3)
8 is the HMO resonance exchange integral and N is the number of formal double bonds possessed by any single structure. The Pauling bond orden are obtained from inspection of the Kekule resonance structures. The method was applied in calculations of resonance effects in electrophile substitutions, Diels-Alder reactions, radical stabilities, ionization potentials, and spectroscopic red shifts. Predictions and correlations were found to be in substantial agreement with results of MO calculations, even for excited state properties. The success of this simple theory in a variety of applications is certainly remarkable. Unsatisfactory aspects of the theory were discussed, and it is interesting that some of the objections that were outlined have been eliminated by more recent work. One objection had to do with the tedium involved in writing 12
/ Journal of Chemical Education
~~~
Vlll
-
were considered to -aive rise to stabilizina exchange - energies. yl refers to resonance between two structures related by~. a permutation of three bonds within a sinale rina, and yz involves the permutation of five bonds wichin two annelated rings, The ratio of y z to y l was obtained empirically from the ratio of the electronic transitions for benzene and azulene. The structures in resonance hybrids were assigned equal weights, and the resonance interactions were enumerated with resonance energies calculated from the formula given in eqn. (4) RE
-
2(n371+ n2r2)+ KSC
(4)
where nl and nz are numbers of resonance interactions of and KSC is the Kekule structvoe .. Y,. - and r.z-. respectively, ture count. Resonance energies, determined by Dewar and de Llano, range from 0.169 eV (azulene) to 5.309 eV (quarterrylene). Corresponding values calculated with eqn. (4) are 0.310 and 5.296 eV. For 29 com~oundsthe average deviation between the two calculations is k0.04 eV (correlation coefficient 0.998). The results indicate that valence-bond theory limited to Kekule structures is a viable concept, but the exact theoretical reasons for the high correlation have not been elucidated. It was also shown that HMO results correlated poorly with either the resonance theory or the SCF-MO values (32). Counting Resonance Structures
The methods used in discovering ways of enumerating resonance structures are to a degree familiar to all chemists. A theory of enumeration is postulated, and the exper-
imental test of the theory is to compare the prediction with experiment. The experimental result is the number of structures obtained by sitting down and drawing all possibilities. If the reader wishes, any of the formulae given below may be checked by this empirical test. G. W. Wheland's Procedure A pioneer of valence-bond theory, G. W. Wheland, described an elegant combinatorial method for obtaining the SC in 1935 (33), a few years after the initial work on valence-bond theory. Actually his method gives the number of cannonical structures for each degree of excitation. In
X
examples of unexcited, mono, di, and triexcited structures are diagramed for naphthalene, and all structures of all excitation types are considered necessary for a complete description of the molecule by valence-bond theory. Associated with each molecule is a polynomial of the form given in eqn. (5). P(z) = ko + k,z k,zZ + ... k.z" (5)
+
The ki are the number of structures of the ith degree of excitation, and z is simply a parameter. Proofs were given for the recursion formulas, eqns. (6) and (7)
These same two compounds will be used to illustrate other methods of counting structures that will follow. In both XI and XI1 the procedure can actually be carried out mentally using the starting kc, structure. Wheland's article (33) also discussed an analogous method for determining the number of structures of any degree of excitation for odd alternant systems. However, the method of LonguetHiggins ( 2 1 ) using the coefficients of a NBMO is so easy to apply that we will not add to the discussion by describing Wheland's approach. Combinatoriai Method of Gordon and Davison (34) In this work a different manner of procedure is required for every different class of compound. The first type considered was an unbranched string of hexagons that could have any number of "kinks." Compound XI consists of a strip of five hexagons with a kink between the third and fourth (or second and third) hexagon. To find the SC of XI, illustrated in XI11
SC
for chains or rings of 2N orbitals respectively. One can see that the number of structures for a particular molecule depends upon the number of structures that can be drawn for smaller fragments of the molecule. Finally, Wheland demonstrated that if a molecular figure, A, differs from a second figure, B, by having one additional line, and if deletion of this line and its two end points divides A into two parts, C and D, then the polynomial for A is given by eqn. (8). (1 - =)PCP, (8) P A = P, If only the number of Kekule structures, given by ko, is desired the procedure is quite simple. From eqn. (6), ko(chains) = -1, and from eqn. (7). kdrings) = 2. Also, ko for a chain of zero atoms is taken to be unity. Then any polycyclic molecule is constructed by drawing a single chain or ring containing all atoms of the molecule, followed by insertion of lines one at a time till the molecule is complete. With each line insertion the SC is obtained from eqn. (8) or its equivalent form eqn. (9).
-
SC
11
-
11
xm one carries out a summation starting a t either terminal hexagon, adding one figure for each hexagon. The numbers start with 2, 3, . . . , one unit being added for each ring. After passing a kinked ring, the quantity added to the previous figure is no longer unity, but is the figure immediately proceeding the kink. An additional example,
XIV
+
'has,
&A,
+ kwokm,
(9)
Two examples are shown in XI and XII.
(3Z&%d?& SC
11
k, = 2
k =2+11)(1)
-
3
&&& 11 = 3
+ 11)(1)
4
k
4
i31(1) = 7
XI
k
7 + ( 4 ) ( 1 ) = 11
SC = 84 XIV
shows how quickly the method can be applied. For compounds of the type given in XII, several general formulas were derived. Benzo[l,lZ]perylene, XII, is classed as a reticulate three tier strip compound, one end indented, of length n = 2. The general formula for such compounds is SC = (n + l)(n 2)(2n 3)/6. This gives S C = 14 in agreement with the previous result. The Gordon and Davison method (34) is interesting but more general and simpler methods can yield the same results. However, one useful point takep from their work is the following. If a molecule can be divided into two or more stable molecular systems for which Kekule structures can be written, and if the sub-systems are joined through positions that belong to only one set of alternant positions in either sub-system, then all joining bonds are single bonds in all structures of the composite molecule. The SC is then the product of the SC's for the snb-systems. SC for zethrene, XV, is therefore (3)=(1) = 9, and that for quaterrylene, XVI, is 34 = 81.
+
+
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It is known from the theory of equations that the constant term in such an expression is the product of the roots of the equation. By eqn. (lo), a, is related to the number of Kekule structures, so one can also obtain the structure count of a n-system by examining the HMO secular equation. Acknowledgment
The financial support of the Robert A. Welch Foundation is greatly appreciated. Several stimulating discussions with Prof. William Leahey, Department of Mathematics, University of Texas at El Paso, are also acknowledged. Literature Cited (1) Pauling, L.. "The Nsmm of the Chemical Bond." 3rd Ed.. Cornell University Ress. Ithaca. N.Y.. 1%0. (2) Wholand. G. W.. "Resonane. in o q s n i e Chemistry," John Wiloy and Sons. New
~~~~~. ~ ~ - - .
",k
WSS.
(3) Pauling, L.. and Wheland, G. W.. J. C h m . Phys. 1.362(1933). (4) Pauling, L.. and Sherman. J., J. C h m . Phys., I. m9 (19331. I51 Whe1and.G. W.. J . Chem. Phys.. 2.474(1934). (6) Who1and.G. W.. J. Chem Phys 3,230(1935). (71 Coulson.C.A..Proc. Roy Sac. Se7.A. 207,91(1951). 18) Coulaon. C. A,, in Thydeal Chemistry. An Advanced Treatise.'' H.), Vol. 5, Academic Press, NewYork, 1970, p p 37683. (9) Oaudol, P.,andDaudel, R., J. Chem Phys., 16.639(1948). (10) 0akley.M.B.. sndKimhall. G. E.. J Chem. Phys.. 17.706(1949). 111) Longuet-Higgms. H. C . J . Chem. Phyr.. 18.265 11950). (12) Dcwar, M. J. S., and Longuet-Higpins, H. C.. Roc. Rgv. Soe, (1952). I131 Pilar, F. L., 'llementary Quantum Chcmiatry." MeGraw-Hill Bmk Co., New York, 1968. p. 517. (14) Streitwieser, A,, Jr., "Molecular Orbital Theory for Organic Chemist," John Wiley and Sons, New Yak, 1961. (15) Clar, E.. "Ammatische KohlenwsssentoNe." Verlag, Julius S~htgcr. Ber.
Ill. 1941. p. 311. 1161 Clar, E., "Polycy& Hydmearbona," Vol. 1, Academic k, New Yak. 1964, chap. 5. (171 Hemdon, W.C., Tefrohdmn, 29,3(l973). (18) R i w . N. V., "Quantum Chemistry," Maemillan Pvbliahing Co., T m t a . 1969. (19) nn i*"d 7;.-. 1vkw1e.T.. Cmal. Chem.Aeto, M.351(1972). 1201 Hjgsai, K.. Baba. H., and h m b a u m , Am. "Quantum Organic Chcmiatry." Wiley-Interscience. NrvYork, 1965, pp. 1620. 121) Paulinz, L.. Bmkruav. L. 0.. and Beseh.. J. Y... J Amer. Chom. Soe.. 57. %705 (19%). (22) Coulson,C. A,,&. Roy. S o t , S e t A. 169,413(1939). . , h r . SCA, n8.n0(1960). (23) ~ m i c k s h a n k ,W. ~ . ~ . , a n d s p a r k ~ , ~ . ~ RO~.SOC., 124) Cruickahank, D. W. J., TeLmhodmn, 17.155 (1962). (251 Ham, N. S.,J. Chem. Phya., 29.1229 (19581. (261 Platt. J. R., in "Handbuck dcr Physik." (Editor S. Flugge), Springer-Vmlag,B m lin, 1 9 6 1 , ~I7bZ8l. ~. (27) Bartell, L. 8..J. Phw. Chsm., 67,1365(1963). (281 Bartell. L. S., Tetrahaddon. W, 13911961). 129) Korafs, E.. Gunfhard, H. H., and Plattner. P. A,. Helv Chim. Alto. 38. 1912 (19651. .(30l I)rwar,M. J.S.,anddaLlano,C., J A m w . Chem. Sm., 91,7PJ(1969). (31) Heaa, B.A., Jr.,sndSehsad,L. J.. JAmer. Chem. Sac. 93,305(1%71). (321 Hemdon, W. C..J Amer. Cham. Soc.. 95.2404 (1973). (331 Whe1and.G. W., J. Chem.Phw., 3.356119351. (34) Gordon, M., and Davison, W. H.T.. J Cham. Phys., W, 428(1952). (35) Stmitai-r. A., Jr., Iewiuia. A,. Fish, R. W.. and Labana, S.. J . Amer. Chem. Sm.. 92.6525(1970i. (38) Wilcax. C. F.. Jr., TelrohdmnLdl., 79511968). r sac., 91, n 3 2 (1969). 137) wibor, C. F., ~ r .J, ~ m e them. (38)Kasfoloyn. P. W.. in "Graph Thew and Thmmtieal Physics;' (Editor: H-. F.1, A c a d e m i c h . N e w Y o r k . 1967.pp.U-108. submitted for (39) Lee. M. A.. Horndon. W. C.. and Phsn. V. T.. Aero Mothem.tie.. p"bli$atio". (401 Cmtkovie. D.,Gutman.I..andm?Mdc,N., Cham. Phys Lett., l6.614(1972). 1411 Graovse. A . Gutman, I.. Trinajatie, N. and Zivkoric, T.. Theor. Chim Acto (Berlin). 26.67 (1972). (42) Cvotkwk, D., Gufman, I.! andTRnaj&, N., Croat. Chem. Aeto. 44.365 (1972). (43) Gutman. I.. andTrinajafic. N., Chem. P b a . Left., 17.535 119721. (40 Caul%on,C.A.. CambridppPhil. Sac. h e . . 46.202(1950). (451 Gimthard. H % H . andRimas, H.,Hdu. Chfm Alto. 39.1&45(1956). (46) Hasoya. H.. Thoor Chim. Acta (Bdin). 25.215 (1972) (47) Hemdon. W. C., and Ellzey, M. L.. J . Comb Moth SP,. B . submitted for publiestion.
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