1. J. Schaad and 8. A. Hess, Jr. Vonderbilt University Nashville, Tennessee 37235
I
I
Hiickel Theory and Aromaticity
After several years in disrepute, the simple Hiickel molecular orbital (HMO) method, when used with proper reference structures, now appears to give accurate predictions of aromaticity ( I ) . To decide whether a molecule is aromatic, one first computes the Huckel a energy in the ordinary way ( Z ) , and then subtracts the energy of a reference structure. The molecule is aromatic, nonaromatic, or antiaromatic depending upon whether this difference is positive, zero, or negative. Two examples will show how this is done, and the remainder of the paper will show why. First consider benzene
structure is obtained by adding the corresponding bond energies from the last column of Table 1. E,,,
=
3(2.0699
+ 0.46601 /3
= 7.6077
0
Subtracting this from the Huckel a energy of 8 B gives a resonance energy (RE) of 0.3923 0,and dividing by the number of a electrons gives a resonance energy per a electron (REPE) of 0.065 8. This is positive, and hence benzene is predicted to be aromatic. Further examples introduce no new principles. Thus fulvene
H*
H
H H
whose Huckel a energy is 8 P. The localized form of this molecule has three single and three double carbon-carbon bonds. These bonds are further classified according to the number of attached hydrogen atoms as in Table 1 where five types of double and three types of single bond are shown. The first digit of the bond type tells whether the bond is double (2) or single (1); the second gives the number of attached hydrogens (0, 1, 2, or 3). We distinguish the double bond with two hydrogens on one carbon (type 22') from that with one hydrogen on each carbon (type 22), hut we do not distinguish between cis and trans attachment of hydrogens. Benzene contains three bonds each of types 22 and 12. The energy of the benzene reference Table 1. Hiickel
7
Bond Energy Terms
is treated in the same way to give REPE
=
1
g(7.47 @ - 2E," - 2E,,"
- E,," - E?,.')
=
-0.00213 This is essentially zero; and fulvene is predicted to be nonaromatic, in agreement with its experimental properties. Historical
Since Huckel introduced his approximation method in 1931 (3), chemists have hoped to use it to distinguish aromatic from nonaromatic molecules. Usually attempts were made to correlate aromaticity with delocalization energy. For a conjugated hydrocarbon with n double bonds in its classical Kekuli. formula, delocalization energy (DE) is defined as the difference between the a energy of the hydrocarbon and the T energy of n ethylene molecules. For example
(E,
=
80) ( E ,
= 8 X 2
0)
Large molecules tend to have more D E than small molecules with graphite having most of all. I t is therefore more reasonable to compare DE per electron rather than total DE. But even when this is done, results are not encouraging. Table 2 shows the order of decreasing D E per electron for a variety of hydrocarbons. Chemists have not agreed
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Table2. Hiickel Delocalization Energy per r Electron (in Unitsof 8 )
naphthalene
0.368
calieene
0.367
simple dsrjve tivea isolated and undergo &ctmphili,: wbstitvtion
..ule"e
0.336
generally eoneider4 aro-
0.333
pmtotype of aromtititr
0.307
methyl deriuative observed spect-opi-
Table 3. Dewar and de LlanoResonance Energy oer r Electron (in eV) benzene naphthalene banzcyclobutadiene ea1i-e azulene fulvalene fv1vene heptalene pentalene cyelobutadiene
well known
aromatic
Table 4. Hess and Schaad Resonance Energy per r Electron (in Units of B) benzene naphthalene ealicene azuiene fv1vene heptalene penta1er.e benzcyclobutadiene fvlvalene cyclobutadiene
matic
a l l y st -196'C.
Van-
ishes when warmed to 100'C isolated, but reactive
-
0.302 0.298
never pmpared
0.280
prepared in very dil"fe solution, very reactive
0.244 0.OW
isolated, but leadive ohserved spec-
troscopicieally at S'K. Vanishes at 36'K.
'See Ref. (19) for a fiat of experimental ref--.
upon a quantitative experimental definition of aromaticity, hut nevertheless it is clear that the DE values of Tahle 2 do not coincide even with qualitative estimates of ammaticity. About the only thing that looks right is the placing of cyclohutadieue a t the bottom of the list. Even this result is not entirely correct since the computed D E of 0 0 says that cyclobutadiene should be as stable as ethylene, whereas it is much less stable. Lin and Krantz ( 4 ) , and indeoendentlv Chapman, McIntosh, and Pacanskv (5), have- generatkd cyciohutadiene by low temperature photolysis of a-pyrone. Cyclohutadiene is so unstable that i t is destroyed by warming to 35'K. Part of this instability is due to ring strain, hut part also appears due to the a electron system (6). Our discussion will he concerned only with the second factor. Another obvious flaw in the list is that henzene should be at the top in place of naphthalene, since one ring of naphthalene can he hydrogenated to leave a more difficultly hydrogenated benzene ring. Benzene certainly should he ahove azulene which can he isomerized to naphthalene (though ring strain may play some part here). Fulvene should he ahove fulvalene. Pentalene should be below heptalene and fulvene. In spite of much experimentation with modified Hiickel and more complex methods, nothing looked promising until some work with the Pariser-Parr-Pople method puhlished by Dewar and de Llano in 1969 (7) and reviewed in this Journal by Baird (8).Their results for the molecules above are shown in Tahle 3. There are points of possible controversy (henzcyclohutadiene is perhaps too high; fulvene and fulvalene should he inverted), hut by and large the order is very much improved. I t therefore seemed reasonable to conclude, as Dewar and de Llano did, that ". . . there no longer seems any point at all in carrying out calculations by less refined procedures, in particular the
HMO method or variants of it" (7). However, a closer look a t their paper shows that not only did they use the more complex Pariser-Parr-Pople method in place of the Hiickel method, hut they also compared their energies, not to that of ethylene, hut to the energy of a hypothetical polyene-like structure. This led us to wonder whether the chanee to a more soohisticated com~utationalnrocedure or the change of reference structure was the more important. I t turned out to he the latter. We found ( 1 ) that the simple Hiickel method, when used with polyenereference structures in the wav shown for henzene and fulvene above gives predictions of aromaticity that are a t least as good as those from the more complicated method. Our results for the same set of compounds are shown in Table 4. The order agrees very well with experiment. v
The Polyene Reference Structure
The origin of the polyene reference structure is a 1965 paper hy Dewar and Gleicher (9) in which they showed that if the a binding energy of linearpolyenes CH2-CH-(CHeCH),-CH=CHx computed h i the Pariser-Parr-Pople method, is plotted against n, a very accurate straight line is obtained. Figure 1 shows that the same result holds for a binding energies from the Hiickel method. In the usual way, a energy is computed in units of 8,taking a p electron on an isolated carbon atom as the zero of energy. The total a energy and r binding energy are therefore equal. I t follows that if one knows the length of the polyene (or equivalently, the number of double and single bonds i t contains) one can
Figure 1. Pi energy of the series CHz = CH-(CH
= CH),-CH
= CH*.
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bonds greater than those for double. All give identical estimates of E, for any molecule. Third, although we have used eight bond types in place of Dewar and de Llano's two, they are not adjusted to fit experimental aromaticity. They are fixed by requiring that equ. (2) hold for acyclics. Predictions of the aromaticity of cyclic compounds will follow automatically without further manipulation of these parameters.
REPE
Aromaticity-A Theoretical Index The r energy of conjugated acyclics is given accurately by adding bond energy terms according to eqn. (2). But this is not true for cyclic compounds, as the example of benzene showed. Had benzene behaved like a polyene, its Hiickel energy would have been E, = 3E,," 3E,," = 7.6071 P instead of the actual value of 8 8. This suggests that the difference between the actual s energy of a molecule and the a energy it would have had if it behaved like an acyclic polyene might be a measure of aromatic character. Accordingly we have defined Resonance Energy per ir Electron ( R E P E ) =
+
(E,
NUMBER O F TI ELECTRONS Figure 2. Resonance energy of the annulenes.
predict accurately the Hiickel s energy. Dewar found (7, 9) that a similar result held for all acyclic conjugated hydrocarbons. That is, for all such molecules T binding energy rr n,E, (1)
+
where nl and nz are the number of single and double bonds in the molecule and El and Ez are constants. At this point the Hiickel results differ a little from Dewar's Pariser-Pan-Pople computations. We found that we could not get a sufficiently accurate estimate of all acyclic s energies using only two bond energy terms as in eqn. (1). Instead we had to distinguish the five types of double and three types of single bonds shown in Tahle 1 with their bond energies. These energies allow an accurate estimate of the Huckel s energy of any acyclic conjugated hydrocarbon using the formula E,
= n,aE,or + n,,E,," +
n,,E,,"
+ n,J2aE,"+ n,,E,," +
n&d
+ n,,.E,,." + n,,E,,"
(2)
where nu is the number of bonds of type ij in the molecule. For example, the estimated s energy of
FCHs
H,C=CH
--
+
TC\ HC=CH,
+
is E. 3 E z d + Ezlr 2Ell* El,. = 9.447 8. This is in very good agreement with E. = 9.446 6 from actual solution of the Huckel secular determinant. Before showing in the next section how Table 1provides reference structures for aromatics, three points should be mentioned in connection with the bond energies. First, they are r energies. Hence, there is substantial a character even for the formally single bonds. Second, it is not possible to construct molecules with an arbitrary number of the eight kinds of bonds. There are in fact two linear relations connecting the eight numbers nlo, rill, . . . m a . This means that two of the bond energy terms in Table 1 must be fixed arbitrarily. Therefore no particular physical significance must be attached to the individual bond energies. Other, equally valid, assignments can be made which give negative bond energy terms or terms for single
.
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- E,*OIN
(3)
where E, = total r energy computed by the Huckel method; Ere, = energy of the molecule calculated by eqn. (2); and N = number of s electrons. Resonance energies in Table 4 were calculated with this formula and, as we have seen, correlate well with experiment. The sign of REPE (in units of 8 ) provides an immediate test of ammaticity R E P E > 0 aromatic compound more stable than a polyene REPE = 0 nonaromatic -compound like a polyene REPE < 0 antiaromatic - compound less stable than a polyene
-
The reference structure to which a cyclic molecule is compared is a hypothetical polyene with the same number of each type of bond as the actual molecule and whose T energy can be calculated additively with eqn. (2). In the case of benzene, the reference is the hypothetical molecule cyclohexatriene. It is not three ethylenes as are used in computing DE; nor is it n-hexatriene. The reference structure takes into account the important, but additive, r contributions of formal single bonds. REPE is a continuous parameter, and one expects no dramatic difference in properties between slightly aromatic and slightly antiaromatic molecules. The annulene series provides an example here. As shown in Figure 2, there is initially a dramatic oscillation between the antiaromatic 4n and the aromatic 4n 2 members. But this oscillation dies out as n increases so that for large n there should be little difference between the two types, with both being practically nonaromatic. For compounds with several Kekul6 structures, the additive energy may differ slightly from one to another. We have always used the average over all structures as reference, but in no case have we found significant energy differences between Kekulk structures. Essentially the same result would be obtained by taking any one as reference. We have found REPE to be a useful index both for alternant and nonalternant hydrocarbons (1, 10-13) as well as for (14, 15) oxygen-, (141 nitrogen-, and (16) sulfur-containing compounds. Figeys (17) and Milun, Sobotka, and TrinajstiE (18) have camed out similar work and have come to similar conclusions.
+
Aromalicity-An Experimental Measure The rough definition of an aromatic compound as one that is especially stable and undergoes electrophilic sub-
stitution rather than addition was sufficient to show that REPE is a better index of ammaticity than DE. But if one wants to make finer distinctions (for example, is calicene really more aromatic than azulene?), a quantitative experimental definition is needed. Diamagnetic susceptihility, nmr chemical shifts, and uniformity of bond length .have all been suggested as measurahle parameten to define ammaticity. We have chosen to follow Dewar (7) again and to use a definition based upon stability. Such a definition is close to the classical idea of aromaticity, and i t also t m s out to be directly connected to REPE. We have shown (19) that, with certain plausible assumptions REPE = (AH. - A H . ( r e f ) ) / N (4) where AH. is the heat of atomization of the compound in question (i.e., the heat required to fragment the compound into separated atoms), N is the number of r electrons, and AH, (ref) is the heat of atomization of the reference structure. AH, and N are directly o b s e ~ a b k hut , AHdref) is not, since the reference structure is a hypothetical molecule. We have shown however (19) that the resonance energy (a property of the r electrons) is linearly related to distortions of the carbon-carbon sigma skeleton. A consequence of this is that if r b i d i n g energy can be expressed in an additive fashion by eqn. (2), a similar equation will hold for AH,. The experimental question to be answered is whether or not heats of atomization for conjugated acyclics can be written in the form of eqn. (2), and hence whether experimental bond energy terms can be measured. If this can he done, AHdref) is expressible in terms of experimental quantities. Unfortunately, not enough thermochemical data are available for conjugated acyclicsto make this test. Nevertheless, we can still proceed as follows. Suppose Huckel predictions of AH. are accurate for those compounds where AH, is known experimentally. Then i t would seem reasonable to assume they are also accurate for conjugated acyclics even though the experimental numbers are unknown. From this, the accuracy of the predicted additivity in eqn. (2) and of the predicted REPE's in Table 4 would also follow.
It might he that the Huckel method can predict the required AH. differences in eqn. (4), and hence REPE's, more accurately than it can individual AH,'s; but in the absence of experimental AH,'s for acyclics, only the more severe test with the individual AHo's can be applied. Consequently, we (19) have used the Huckel method to compute AH.'s for a set of compounds previously studied by Dewar and de Llano (7). An average error of less than 0.1% was found for calculated AHo's which is slightly, but not simificantlv. better than Dewar and de Llano's Pari s e r - ~ & - ~ o ~ l e ~ r ~ sWe u l t have s . also shown that the accuracy of our REPE's is *0.004 B and that B = -32.74 kcall mole. Note we have not changed the Huckel method itself; it was used in its original and simplest form. Only the reference, against which a computed r energy is compared, has been modified. I t is gratifying that the simple HMO method does, after all, appear to contain the key to ammaticity.
Literature Cited (1) Has, B.A..Jr., and Scbaad, L.I.,J.Amp,. Chem. Soc., 93,305 (197,). (2) Streitw%asr, A . Jr.. "Molecular Orbital Th-ry for Oqagsnic Chemists." John Wilcy&Som.Ine.,NcvYork,N.Y., 151. (3) Hackel. E.,2.Physik, 10.204 (1931). (4) Li., C.Y.. and Krant..A.. C k m . comm.n. 1111 (1912). (5) Chapman, 0. L.,Mclnfoah, C. L., and Paeamky, J., J. Amer Chrm. Soc.. 95,614 (1973). ( 6 ) B ~ l o u ,R., Grubbe. R.. md Muraharhi, S. I., J Amer Chem. Soe., 92, 4139 (1970). (7) ( a ) Deiuar, M. J . S.. and de Llano. C.. J Amer Chem. Sor, 91, 789 (1969): (b) de Llano. C., PhD DiuenaLion,TheUni~~nityofToxaa. 1968. (8) Bsird, N.C.. J. CHEM.EDUC..18.509 (19711. (9) Oarar, M. J. S., and Gleicher, G. J.,J Am*,. Chem. Sor., 81,692(1965). J A m e r Chsm Soc.. 93.2113(1971). (10) Hess.B.A.,Jr.,andSchsad.L.J., (11) Has, B. A., Jr., and Schasd, L. J.. J Org. Chem., 36, 3418 (1971); 37, 4179 ,,w1),
(12) ~&~'i;:A..~~.,andsehsad,~. J., TetmhedmnLsfl., 17i19111;5L13(19721. (13) Hess, 9. A,, Jr., andSehasd.L. J.,Alrst. J. Chsm.. 25.2211 (1972). (141 Has, H. A., Jr., Sehaad, L. J.. and Holyoke, C. W., Jr., Tefmhedmn, 28, 3656 119791 ,. ... ,. (15)'~pse.H. A., Jr., Schaad, L. J., and Holyoke, C. W. Jr., Telmh~dmn,28, 5299 119721. . -, (16) Roan, 9. A..Jr., andschaad, L. J., J Amar Chrm. Soc., 95.3907 (1973). (17) Figem, H. P.,Tetmhrdmn. Z6.5225 (1970). (18) Mdun, M., Sohotha. Z., andT%ajstiP. N.. J Or#. Chem.. 31,139(1972) (19) Schaad, L.J., and H w , 9. A,. Jr.,JAmer Chpm S o c . 94.3068 (1972). ~
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