6631
Resonance Theory. V. Resonance Energies of Benzenoid and Nonbenzenoid x Systems William C. Herndon* and M . Lawrence Ellzey, Jr. Contribution f r o m the Department of Chemistry, University of Texas at El Paso, El Paso, Texas 79968. Received April 27, 1974 Abstract: Structure-resonancetheory for r-molecular systems based solely on covalent KekulC structures is justified phenomenologically and by reference to recent theoretical work. The idea of antiaromaticity is shown to be a logical extension of the theory, and a concept of a local ring aromaticity or antiaromaticity is quantitatively defined. Estimates of resonance stabilization energies are substantially lower than predictions based on Hiickel M O theory. The resonance theory results agree with those from SCF-LCAO-MO calculations.
T
he recognition of extremely simple algorisms for counting KekulC structures’ and their permutations induced us to test a semiempirical quantum theory with a basis of KekulC structure functions. The mathematical equivalencies3-j between Hiickel molecular orbital (HMO) and valence bond (VB) theories for the benzenoid class of hydrocarbons led us t o expect a correspondence of our results t o previously known H M O quantities. The results2 were surprising in that it was found that calculated resonance energies, RE, for an extensive series did not correlate well with HMO delocalization energies (correlation coefficient, 0.493 for resonance energy per electron, REPE). Instead there was a congruity with resonance energy values obtained from SCF-LCAO-MO calculations6 (correlation coefficient, REPE, 0.991). The approach that we use is essentially a quantification of the structural resonance theory traditionally applied t o structure-reactivity problems in organic chemistry.’ In this paper we try to provide some justification for our procedures which will be described in more detail than in the previous communication. Calculations of resonance energies of several classes of aromatic compounds will then be presented, and comparisons with previous MO results and experimental properties will be delineated. We emphasize throughout that our computational procedure is so easily carried out and leads to such sensible results that it should be the method of choice for calculating resonance energies. Applications to the estimation of heats of formation8 and carcinogenic activitiesg of benzenoid hydrocarbons have already given good results, and a description of bond-order relationships will appear in a following article. ( I ) W. C. Herndon, Tetrahedron, 28, 3675 (1972); 29, 3 (1973); J . Chem. Educ., 51, 10 (1974). (2) W. C. Herndon, J . Amer. Chem. SOC.,95,2404 (1973). (3) H. C. Longuet-Higgins, J . Chem. Phq’s., 18, 265, 275, 283 (1950); M. J. S. Dewar and H. C. Longuet-Higgins, Proc. Roy. Soc., Ser. A , 214, 482 (1952). (4) N. S. Ham and I i
(1)
tions, the c f are coefficients, the index u indicates the type of electron distribution associated with the function, and the subscript i numbers the function. After selecting functions, the coefficients and eigenvalues can be evaluated by solving eq 2, where H is the Hamiltonian
( H - ES)C
=
0
(2)
matrix and S is the overlap matrix. A conventional approximation is to assume zero overlap of the wave functions so that off-diagonal elements in the matrix involve Hamiltonian integrals only. A significant simplification in obtaining a ground-state eigenvalue is also provided if one assumes a wave function consisting of equal contributions from the class of functions corresponding to KekulC structures. Then the eigenvalue or resonance energy is calculated from
E
=
(2/K.S.)(ZHtj)
(3)
The matrix elements H i j between the structure functions can be evaluated theoretically after superimposing Rumer-Pauling diagrams, corresponding to the electronic arrangements of $ iand $ j . We chose, however, to determine these integrals from spectroscopic data by a method described by Simpson’* which follows the ideas of P a ~ l i n g ’and ~ Forster.14 Simpson postulates undefined, polyelectronic wave functions whose squares have the transformation properties of structures. The set of these structure functions forms a basis for a reducible representation of the pertinent molecular point group. Different states are represented as appropriate symmetry adapted linear combinations of structure functions, equivalent to appropriate combinations of VB structures. Particular experimentally determined term values are then taken as elements of a diagonalized square matrix, the order of (11) L. Pauline. J . Chem. Phvs.. 1. 280 (1933): J. H. Van Fleck and A.‘Sherman, R e l ’ M o d . Phys;, i,167 i1935j. (12) W. T. Simpson, J. Amer. Chem. SOC.,75, 597 (1953); W. T. Simpson and C. W. Looney, ibid., 76, 6285 (1954); C. W. Looney and W. T. Simpson, ibid., 76, 6293 (1954); W. T. Simpson, ibid., 78, 3585 (1956). (13) L. Pauling, Proc. Nat. Acad. Sci. U.S., 25, 577(1939). (14) Th. Forster, Z . Phjs. Chem., Abt. E , 41,287 (1938). ~
I ,
Herndon, Ellzey 1 Resonance Energies of
P
Systems
6632
which must correspond to the number of basis structures. The diagonal matrix is orthogonally transformed to nondiagonal form, where base vectors in the transformed system are interpreted to represent isoenergetic structures. The application of this approach in classifying and understanding the spectra of aromatic dyes was discussed in detail. l 2 Apropos of the present work, it was found that first electronic transitions of benzene (lAlg + IBZu,4.89 eV), naphthalene (lAg + IBaU,3.97 eV), and azulene (’A1 ‘B2, 1.79 eV) were assigned consistently with correct polarization when bases of Kekult structures were used. The off-diagonal elements of the transformed matrix are the H , , that can be substituted in eq 3 to give the ground-state resonance energy. Examination of all KekulC structures for benzene and for azulene indicates that in effect the H,, are integrals that result in permutations of pairs of ir electrons over the u bond framework --f
exceedingly large number of canonical nonionic structures can be written for even relatively small ir-molecular systems (anthracene, 429 structures; benzanthracene, 4862; dibenzanthracene, 58,786). Several workers have shown that for most aromatic molecules the total weight of long-bond structures dominates over the total weight of KekulC structures.’* Coulson has emphatically stated that it is not appropriate to use the VB resonance method when there are more than about ten atoms in the ir system.lg Second, a large body of theoretical evidence is accumulating on the importance of including ionic structures in VB calculations. 2o A recent ab initio VB calculation on benzene indicated that covalent KekulC and Dewar structures 3 are relatively unimportant in the ground-state wave function as compared to singly polar structures 4. The
3 1
4
2
of an aromatic molecule. Three pairs of electrons in a single ring are permuted by yl, and five pairs of electrons in two annelated rings are permuted by y2. The ratio of y2 to y1 is given by the ratio of the electronic transitions given above for benzene and azulene, y2/y1 = 0.37. This result is in close agreement with a value of 0.36 which is obtained from VB superposition diagrams with values of Coulomb and exchange integrals calculated by Coulson and Dixon.’j Also, anticipating later results, the best correlative value of the ratio of y2/y1is 0.40 (-yl = 0.841 eV) by comparison with a large number of SCF-LCAO-M06,16 calculations of resonance energies. The SCF method, which we will call the Dewar-de Llano method,6 is a variable bondlength semiempirical approach parameterized with experimental thermodynamic values, and it provides very accurate estimates of heats of formation and very reasonable values for resonance energies. The resonance theory results for benzenoid hydrocarbons2 show that it is not necessary t o include the effect of integrals that result in permutations of larger than five pairs of electrons in order to obtain resonance energies consonant with the previous theoretical work. However, two- and four-electron pair permutations are demonstrated to have a considerable effect in assigning resonance energies, but a discussion of their relative values will be deferred to a later section following the detailed outline of the computational procedures. Pragmatically, one seems justified in using the small limited set of KekulC structures as basis functions for a calculation of resonance energy. One notes that many practitioners of VB theory have not apologized for choosing a small set of “reasonable” structures from among all possible canonical structures and ionic forms that could be drawn.I7 However, there are two reasons why more justification may be necessary for the compounds discussed in the present paper. First, an (15) C. A. Coulson and W. T. Dixon, Tetrahedron, 17, 215 (1962). (16) C. J. Gleicher, D. N . Newkirk, and J. C. Arnold, J. Amer. Chem. SOC., 95,2526 (1973). (17) Examples: H. M. McConnell and H. H. Dearman, J . Chem. Phj,s., 28, 51 (1958); J. C. Schug, T. H. Brown, and M. Karplus, ibid., 35, 1873 (1961); W. J. Van der Hart, J. J. C. Mulder, and L. J. Oosterhoti, J . Amer. Chem. SOC., 94, 5724 (1972).
Journal of the American Chemical Societj.
1 96:21
number of these polar forms is also much larger than the number of nonionic structures. The difficulties outlined in the previous paragraph should not be ignored, but they are not really consequential in the context of the present calculations. It is important to remember that the theory used here is a parameterized theory that uses VB structures only as representations of squares of undefined, manyelectron wave functions. The assumption is that the transformation properties of the underlying wave functions are the same as those of the structure representations. Simpson points out l 2 that for benzene the sum of the unknown wave function must transform like AIg, and the difference like BZu,and he states that attempts t o draw such entities will result in drawings which closely resemble KekulC structures. Quantitative realizations of these “drawings” are found in recent calculations by Paldus, Cizek, and Sengupta. Their separated-pair localized geminal calculations using a Pariser-ParrPople-type Hamiltonian gave very good ground- and first excited-state energies, and the wave functions used are exact analogs of KekulC structures. Several years ago Dewar and SchmeisingS3suggested that the individual KekulC structures have a more profound significance than is usually ascribed to the canonical structures of VB theory. The views of Simpson” and the calculations cited abovet2 show how this apparent anomaly can be resolved. Our own results indicate that, with empirical parameterization of matrix elements, the use of Kekuli structures alone yields resonance energies of SCF-LCAO-MO quality. (18) A. Pullman, Ann. Chim. (Paris), 2, 5 (1947); C. Daudel and R. Daudel, J . Chem. PhJ’s.,16, 639 (1948). (19) C. A. Coulson, Proc. Roy. Soc., Ser. A , 207, 91 (1951); C. A. Coulson in “Physical Chemistry, An Advanced Treatise,” Vol. 5, Academic Press, New York, N. Y., 1970, p 379. (20) A. L. Sklar, J . Chem. Ph),s., 5, 669 (1937); D. P. Craig, Proc. RoJ,. SOC.,Ser. A , 200, 401 (1950); R. S. Berry, J . Chem. Phys., 30, 936 (1949). \ _ _ _ _ ,-1. , C. - Schue. ibid.. 42. 2547 (1965); M. Craig and R. S. Berry, J . A m e r . Chem. Socy: 89, 2801 (1967). (21) J. M. Norbeck and G. A. Gallup, J . Amer. Chem. SOC.,95, 4661 (1973). (22) J. Paldus, J. Cizek, and S . Sengupta, J . Chem. Phys., 55, 2452 (1971); S . Sengupta, J. Paldus, and J. Cizek, I n t . J . Quanrum Chem., Symp., No.6 , 153 (1972). (23) M. J. S . Dewar and H. N. Schmeising, Tefrahedron, 11, 96 (1960).
October 16, I974
-4 6633
Computational Method The exclusive use of KekulC structures for benzenoid compounds in a structure-resonance theory is justified in the previous section. To implement the theory one needs t o enumerate the number of KekulC structures and the numbers of matrix elements that convert one structure into another. We will refer t o these matrix elements (yl and y2)as permutation integrals or resonance integrals, and they are pictorially defined in 1 and 2. Of course one can simply draw all structures and count all required permutations. To do this we find it convenient t o construct a graph, vertices of which represent Kekult structures, and edges of which are weighted by the values of the permutation integrals. The triangular graph shown for naphthalene in 5 is a
\
A
/
Y:! 5
graph of this type, and it bears a superficial resemblance to the HMO graph of the cyclopropenyl system. One can imagine eigenfunctions of the graph in 5, each eigenfunction corresponding to a particular eigenstate. The graph eigenfunctions transform according to the irreducible representations of state symmetry species. The ground-state graph eigenfunction for naphthalene (A,) therefore has no nodes and a good approximation t o its resonance energy is given by eq 3.24 Using yl = 0.841 eV and y2/y1 = 0.400 the resonance energy is (2/3)(2yl y2) = 1 . 6 0 0 ~ (1.346 ~ eV), which is close to the value of 1.323 eV computed by Dewar and de Llano.6 The average deviation of the resonance method results from the LCAO-SCF-MO values is iO.037 eV for all of the benzenoid molecules that we have examined. For larger benzenoid molecules, the writing of resonance structures can become tedious. It is easier t o use the graph-theoretical concept of the “structure count” (SC) described in recent papers.’ The SC method requires one to delete a vertex from the graph of the aromatic molecule and then t o write the nonarbitrary vertex coefficients (smallest coefficient unity) that sum t o zero around every vertex in the residual graph. Most readers will recognize these coefficients as the unnormalized coefficients of a nonbonding molecular orbital for the odd residual system.25 The sum of the absolute value of the coefficients adjacent to the deleted vertex is the SC, i.e., the number of KekulC structures that can be drawn for the original molecule. Then the number of y1permutations for each ring in the molecule is the SC for the residual molecule with that particular ring excised from the structure. Similarly, y2’s are enumerated by deleting adjacent rings two at a time and summing the SC’s for the residual systems. The whole procedure is economically carried out on three drawings of the molecular graph as illustrated in 6
+
(24) W. C. Herndon and E. Silber, J . Chem. Educ., 48,502 (1971). ( 2 5 ) General procedures to obtain nonbonding orbital coefficients are described in ref 1 and in T. Zivkovif, Croat. Chem. Acta, 44, 351 (1972).
1
sc = 13
.VT2 = 7
8 y , = 21 6
for dibenz[a,c]anthracene. The resonance energy of dibenz[a,c]anthracene is therefore (2/13)(21y1 3- 7y.4 = 3 . 6 6 2 (3.079 ~ ~ eV). The SCF result6is 3.058 eV. Antiaromaticity Many of the molecular structures that we wish to examine in this paper can be at least partially represented by KekulC structures that are related by permutations of even numbers of pairs of electrons. Cyclobutadiene (7) and pentalene (8) are two structures
8
7
of this kind, and larger molecules may incorporate either one of these structures or both types as part of the molecular framework. The permutation integrals characteristic of resonance between the structures in 7 and 8 will be called w1 and w 2 ,respectively. The properties of these molecules show that they have fundamentally different characters from the benzenoid hydrocarbons. There is no doubt that benzenoid hydrocarbons are resonance stabilized, where we accept the common definition that a compound is resonance stabilized if cyclic delocalization of 7r electrons stabilizes it relative to an open-chain model compound.2R Benzenoid compounds also would be classified as aromatic compounds by any of the several criteria that have been suggested to define aromati~ity.~’In contrast, the cyclobutadiene structure has been suggested to be an “antiaromatic” structure, 28 with 7r-electron energy higher than that of two isolated or linearly conjugated double bonds. The idea is an outgrowth of the longrecognized 4n 2 ( n = integer) Hiickel rule for aromaticity used to explain the relative stability of the aromatic sextet. 29 MO calculations confirm the idea of antiaromaticity for cyclobutadiene. 3 0 - - 3 2 The Dewar-de Llano method6 gives a destabilization of -0.78 eV attributable t o cyclic delocalization of 7r electrons in cyclobutadiene. 3 2 Interestingly, this destabilizing energy is obtained after distortion of the square-planar molecule to an extreme rectangular form with essentially alternating double and single bonds. This shows that it is not possible to
+
(26) M. J. S. Dewar and G. J. Gleicher, J . Amer. Chem. SOC.,87, 692 (1965). (27) For recent reviews see (a) A. J. Jones, Rer. Pure Appl. Chem., 18, 253 (1968); (b) G. M. Badger, “Aromatic Character and Aromaticity,” Cambridge University Press, Cambridge, 1969; (c) “Aromaticity, Pseudo-Aromaticity, Anti-Aromaticity,” E. D. Bergmann and B. Pullman, Ed., Israel Academy of Sciences and Humanities, Jerusalem, 1971; (d) I. Agranat, M T P Itit. Rec. Sci. O r g . Chem., Ser. One, 3, 139 (1973). (28) R. Breslow, Accounts Chem. Res., 6, 393 (1973). (29) A. Streitwieser, Jr., “Molecular Orbital Theory for Organic Chemists,” Wiley, New York, N. Y., 1961, Chapter 10. (30) N. L. Allinger and J. C. Thai, Theor. Chim. Acta, 12, 29 (1968). (31) R. J. Buenker and S. D. Peyerimhoff, J . Chem. Phys., 48, 354 (~~. 1 968). . ~ ,
(32) M. J. S. Dewar, M. C. Kohn, and N. Trinajstif, J . Amer. Chem. SOC., 93, 3437 (1971).
Herndon, Eflzey
Resonance Energies of H
Systems
6634
avoid this consequence of antiaromaticity by distortion to a formally localized structure. In the context of the structure-resonance theory, this could mean that one cannot avoid introducing w1 and w2 terms by judiciously selecting a subset of Kekult structures to represent a particular molecule. In the past, selection of particular KekulC structures has not been considered to be undesirable. For example, the structure of biphenylene has several times been postulated to be characterized by a single resonance structure 9a or the set of three structures with structures 9d and 9e excluded. 33-35
“(JfJ
reactive. Two other aspects of molecular structure consistent with destabilization and distortion of 4n 7r-electron systems are the nonplanar alternating structure of c y c l ~ o c t a t e t r a e n eand ~ ~ the alternation of bond lengths in [16]annulene from X-ray data.45 Lastly, Breslow has described the results of some very elegant thermodynamic experiments designed t o delimit the magnitude of antiaromaticity. 28,46 From the observed oxidation potentials of 12a (- 1.15 eV) and 13a (- 1.67 eV), he argued that cyclobutadiene con0-
/
b
a
0-
C
0 b
a
12
d
0-
e
0
9
Another well-known peculiarity of 4n 7r-electron systems is their susceptibility to pseudo-Jahn-Teller distortions from high symmetries to lower symmetries of the nuclear framework.j6 Both recent VB15 37 38 and MO j 2 38-4 calculations agree that the effect should manifest itself in cyclobutadiene and pentalene, with cyclobutadiene having a rectangular ( D 2 J singlet state and pentalene having alternating bonds and CZhsymmetry. Bond fixation in 7 and 8 is also supported by experimental facts. For example, 1,2-diphenylcyclobutadiene is known t o react with weak dienophiles to yield a single Diels-Alder adduct (from loa) and with C, H CH
>= > Ci H
C, H
a
b 10
more reactive dienophiles to give two adducts characteristic of both 10a and lob, indicating that the cyclobutadiene exists as an equilibrating mixture of the two highly reactive and unstable, singlet isomers. 4 2 Pentalene is not known, but the phenyl-substituted derivative 11 has been ~ y n t h e s i z e d ,and ~ ~ it is also unstable and CH \
C,H I
C3;
GHi
11
(33) W. Baker, J . F. W. McOmie, D. R. Preston, and V. Rogers, J . Chem. SOC.,414 (1960). (34) T. C. W. Mak and J. Trotter, J . Chem. SOC.,1 (1962). (35) J. I
