JOURNAL O F THE AMERICAN CHEMICAL SOCIETY Regivered in U S Parent Offlce
0 Copvrighr, 1974 bJ the American Chemical Sociely
V O L U M E96, N U M B E R25
D E C E M B E R1 1 , 1 9 7 4
Resonance Theory. VI. Bond Orders William C. Herndon Contribution from the Department of Chemistry, University of Texas at El Paso, El Paso, Texas 79968. Received May 23, 1974
Abstract: Graph-theoretical and MO formulations of Pauling bond orders are outlined. Negative bond orders and bond orders greater than unity are defined and discussed. The use of Pauling bond orders to correlate various types of chemical and physical properties is justified as being tantamount to using the results of SCF-LCAO-MO calculations. Bond lengths and nmr ortho coupling constants in benzenoid systems can be accurately predicted using functions linear in Pauling bond orders. Of most significance is the fact that the bond-order relationships can be extended to include nonalternant hydrocarbons and annulenes without loss of correlative abilities.
The interrelations of certain physical properties and theoretical bond orders in benzenoid aromatic hydrocarbons are well-established. As examples, one finds that there are linear correlations of Coulson ir bond orders] (Hiickel molecular orbital method, HMO) with experimental bond length^^-^ and with nmr spin-spin coupling constants of ortho hydrogen atoms, J HH0rtho.7-12 A theoretical rationale for empirical bond order-bond length relationships has been outlined by Salem,13,14and the linear ?r bond orderJ H Hrelations are probably a consequence of dependence of J H Hon u orbital overlap, and hence on bond length.i5-1S More refined MO methods that have been used to correlate bond lengths and coupling constants d o not lead to any appreciable better agreement of calculated and experimental v a I u e ~ . ’ ~ ~ ~ ~ - ~ ~ The applications of valence bond (VB) theory are less numerous and consider fewer systems. The VB theory of spinspin coupling has been described in d e t a i l l ’ ~but ~ ~ has only been applied to benzene and methyl-substituted derivatives.” The formulation of a VB bond order-bond length relation actually antedates M O treatment^,"^^^ but again has only been applied to a few benzenoid hydrocarb o n ~ . ~ - ~ Nevertheless, ,*~-~’ it has been pointed out for particular molecules that Pauling bond orders are as successful in correlating bond lengths as any other kind of theoretical method. Recently, it was demonstrated that a parameterized structure-resonance theory that uses only KekulC struct u r e ~ ~gives ) . ~resonance ~ energies for many diverse kinds of ir molecules that are in excellent agreement with the results of variable bond length SCF-LCAO-M035 calculations. Also, there is almost perfect correlation of resonance theory and SCF localization energies with Pauling bond orders.36 These factors, along with extreme ease of calculation, mandate a complete comparison of Pauling bond orders with experimental bond lengths and coupling constants. A main
purpose is to show how the use of Pauling bond orders to correlate structural data can be extended to include x structures other than benzenoid hydrocarbons.
Calculations of Pauling Bond Orders The least efficient way of finding Pauling bond orders is to draw all KekulC structures, finally obtaining the ratio of structures in which a bond is double to the total number of structures. For alternant benzenoid systems many different techniques have been recently s ~ m m a r i z e d . ~The ’ quickest method is to delete a vertex from the molecular graph of the ir system under consideration, obtaining an odd alternant graph for which one can write a nonarbitrary set of coefficients that sum to zero around every vertex in the graph. The reader will recognize these coefficients as the unnormalized eigenvectors of a nonbonding (NB) MO for the deleted vertex system.38 The sum of the coefficients a t vertices adjacent to the deleted vertex is the structure count (SC)39 for the parent system. Furthermore, these coefficients also enumerate the number of Kekul& structures in the parent system in which the bond to the deleted vertex position is double.40 Graphs of deleted vertex systems of benz[a]anthracene are given in 1. The Pauling bond orders deter-
n
s36332
n I
mined from the N B M O coefficients are also shown, and all 7605
7606 remaining bond orders in the molecule follow because the sum of Pauling bond orders around any vertex must equal unity. One additional way of determining Pauling bond orders is of interest. In H M O theory the Coulson bond order1 is defined according to eq 1, where the sum is over j occupied f
levels, n, gives the number of electrons in level j , and .c, and c,? are eigenvectors. Chemical intuition suggests that the quantity given in eq 1 should be a measure of electronic charge in a bonding region. However, the Coulson bond order cannot be considered as analogous to a bond population. The sum of all bond populations is the total number of electrons, whereas the sum of all Coulson bond orders is the total R energy of the delocalized R system. H a m and Ruedenberg4I state that the appropriate theoretical quantities for comparison with bond lengths are bond populations. The Pauling bond order is analogous to the bond population in the above regard, and it is interesting that Pauling bond orders for all alternant systems can also be expressed within the H M O theory as shown in eq 2.41 The partial
with high accuracy by the resulting linear eq 3. A linear
-
(31 equation using Coulson HMO bond orders fits the data with average deviations three times larger. In addition, the bond length in graphite (exptl. 1.421 8,47)is calculated to be 1.422 8, (pp = 0.333) by eq 3 , whereas the Coulson bond-order equation ( p c = 0.535) gives 1.438 8,. For classical cyclic and acyclic olefins that can be represented by only a single KekulC structure, the Pauling bondorder method makes a very straightforward prediction that all single bonds should be 1.464 8, in length and all double bonds 1.339 8,.Relevant experimental bond lengths are listed i n Table 11, and the differences between calculated and drs = 1 . 4 6 4
0. 125fiTSp
Table 11. Single and Double Bonds in Classical Polyenes (A)
Compd
Bond
Bond length" (exutl)
Cyclopentadiene
a b
1 .342b I , 469b
-0,003 -0,005
1,3-Cyclohexadiene
a b
1 . 346c 1 . 467c
-0.007 -0,003
Hexatriene
a b
1 . 337d 1 .457d 1 . 367d 1 ,347e 1 .476e 1 . 3406 1 . 462e 1.356f 1.446j 1,356f
0,002 0.007 0.028 -0.008 -0,012 -0,001 0.002 -0,017 0.018 -0.017
1 . 340d 1 ,475d
-0.001 0.010
Q
Calcd - exptl
f
Coulson bond order of each level is weighted by the inverse of the eigenvalue so that less stable MO levels contribute to the bond order more than the lower levels. The point is that tables of H M O eigenvalues and eigenvectors can therefore be used to determine exact Pauling bond orders. The Pauling bond order has some of the characteristics of a bond population but also bears some resemblance to the Coulson bond order, although eq 1 and 2 show there cannot be an exact linear correspondence. When one applies either of the two Pauling bond order methods described to R systems containing rings of 4n ( n = integer) vertices, one finds that some bonds are found to have negative Pauling bond orders, and others have bond orders that are greater than unity. Neither of these two concepts have been discussed previously, although the occurrence of negative bond orders could be related to the fact that estimated resonance energies for cyclobutadienoid systems are negative, indicating antiaromatic character rather than a resonance stabilized ~ y s t e m . A~ fuller ~ - ~ ~discussion will be given in a later section. Polyenes and Benzenoid Hydrocarbons The accurate optimum experimental values of bond lengths listed by Allinger and Sprague for benzene, ethylene, and butadiene were chosen to define a Pauling bond order-bond length r e l a t i ~ n s h i p .The ~ ~ bond orders and bond lengths are listed in Table I, and the data are fitted Table I. Experimental and Calculated Bond Lengths Compd Benzene Ethylene Butadiene Butadiene
Bond order 0 1 1 0
5 0 0
0
-Bond
lengthExutl
1 1 1 1
397a 337b 343c 466((1 464d)
Calcd 1 1 1 1
401 339 339 464
(A) Calcd - exptl 0 004
0 002 -0 004 -0 002
A. Langseth and B. P. Stoicheff. Caji. J . Phys., 34, 350 (1956). L. S. Bartell. E. A. Roth. C. D. Hollowell. K. Kuchitsu, and J. E. Young. Jr.. J . Cliern. Phjs., 42, 2683 (1965). W. Haugen and M. Traetteberg, Acfa Chem. Scand., 20, 1726 (1966). A . R . H. Cole. G. M. Mohay, and C. A. Osbourne, Specfrochin?.Acta. Part A , 23, 909 (1967).
m
C
Dimethvfulvene
a b C
Cycloheptatriene
. 0,
Cyclooctatetraene
0.
d a b C
a b
-
L. H. a Electron diffraction values except for b (microwave). Scharpen and V. W. Laurie, J . Clrem. Phys., 43, 2765 (1965). Averages from three investigations: G. Dallinga and L. H. Toneman, J . Mol. Struct.. 1, I1 (1967); M. Traetteberg, Acta Chem. Scarid., 22. 2305 (1968); 0. Oberhammer and S. Bauer, J . Amer. Chem. SOC.,91, 10 (1969). M. Traetteberg. Acta Cliem. Scnrrd., 20. 1724 (1966): 22. 628 (1968). e J . F. Chiang and S. H. Bauer, J . Amer. Clrem. SOC., 92. 261 (1970). M. Traetteberg, ibid..86, 4625 (1964).
experimental values are given in the last column. Only the central double bond in hexatriene and the three bonds of cycloheptatriene show a deviation larger than the probable experimental error. The hexatriene molecule is quite flexible and it is possible that the experimental value of the central double bond is lengthened by torsional motions.25 The cycloheptatriene results cannot be rationalized, but the near constancy of double bonds and single bonds in all of the other olefins in Tables I and I1 leads one to suspect that the experimental results may be incorrect. The bond lengths and Pauling bond orders of benzenoid hydrocarbons are listed in Table 111. Also listed are average errors in the experimental bond lengths as given by the respective investigators and average differences of calculated and experimental bond lengths as determined with eq 3. Every benzenoid compound listed in the literature with quoted experimental errors of 0.010 A or less was included in the table. Some of the larger structures illustrate a very useful aspect of Pauling bond orders, in that any interested
Journal of the American Chemical Society / 96:25 / December 1I , 1974
1607 person can immediately check the calculated values. This is not possible for even the simplest of MO calculations. Using eq 3, the average deviation of calculated and experimental values is f0.009 A, very slightly larger than the average estimated experimental error of f0.008 A. Presumably, a statistical fit of the bond lengths to Pauling bond orders would have yielded a n even closer correlation, but the approach to using a few standard molecules has a particular advantage. If the correlation of data is generally good, as in the present work, one can then consider the exceptions as special cases for which ad hoc explanations may be necessary. For example, there are 25 bonds listed in Table 111 which have bond orders larger than 0.500, and 19 of the calculated bond lengths are larger than the X-ray values. These are also the bonds where the calculated values have the largest errors, so if systematic experimental errors could be discovered characteristic of bonds with high bond orders, a great deal of the already small discrepancies could be removed. An examination of the available neutron and X-ray diffraction data48,49suggests the possibility that bond lengths in bonds of high order are underestimated by X-ray diffraction. This is understandable since neutrons are scattered by atomic nuclei, whereas with X-rays the center of electronic charge clouds are determined. One can easily show that the shortening effect in the X-ray experiment should be an approximately linear function of bond order50 with the correction in the experimental value being +0.019 8, for a bond order of 0.800. Calculated bond lengths would then lie well within the limits of the experimental values. There is a variation of bond lengths found in Table 111 that could indicate a severe'limitation of the use of Pauling bond orders to correlate bond lengths. In the worst case, bonds with bond order of 0.250 range in length from 1.420 (anthracene) to 1.468 8, (chrysene). The average range for each bond order is ca. 0.02 8, which is two to three times as large as the quoted experimental errors in Table 111. However, I think it probable that experimental redeterminations of bond lengths in particular molecules might resolve these discrepancies. One notes that many of the supposed anomalies in bond order-bond length relationships have vanished when more accurate experimental data have become available. Bonds a and h in phenanthrene formerly were given as 1.457 and 1.390 A, respectively,j' values that were always troublesome to theorists. Bond f i n pyrene was always calculated to be 0.02 to 0.03 8, larger than the experimental value,49 and bond e in perylene was always estimated by MO calculations to be much shorter than the length of 1 S O 8, from early data.52 For the latter case, a suggestion that overcrowding of 1,g-phenanthroid type hydrogen atoms would be relieved by a carbon-carbon extension of 0.03 to 0.04 8, was widely a c ~ e p t e d . ~ , ~This . j 3 explanation is still advanced for the very long bonds (1.53 A) found in quarterr ~ l e n e11,~ so ~ a new determination of quarterrylene bond 153'4
r/
/
I1
lengths may be in order. Recent force-field calculations generally agree that strain is relieved in aromatic systems by small adjustments in bond angles and out-of-plane deform a t i o n ~ , rather ~ ~ , ~than ~ by bond length adjustments. The results given in Table 111 support this viewpoint. T h e empirical structure-resonance theory mentioned in
the introductory ~ e c t i o nallows ~ ~ , ~one ~ to assign weights to the individual Kekul6 structures, the weights being related to the eigenvectors of the Hamiltonian matrix. The results of such a calculation are illustrated in 111, weights in paren-
( 0 315)
(0.371)
( 0 315)
111
theses. Resulting calculated bond lengths differ from those given in Table I11 by an average of 0.004 A, some deviations negative and some positive, so that the average difference in experimental and calculated bond lengths remains approximately the same. With larger aromatic A systems the weights assigned by resonance theory calculations are even more uniform, so that one concludes that the Pauling bond order based on equivalent contributions of KekuE structures is an adequate description for predictive purposes. In a few small nonalternant systems to be discussed, the relative weights of structures seems to be a more important factor. It is not necessary to present a tabulation of bond orders, coupling constant data, and calculated coupling constants. The extensive previous work on the mutual correlations of Coulson bond orders with bond lengths and coupling constants,> I 2 in conjunction with the Pauling bond order-bond length relationship illustrated by the data in Table 111, makes it apparent that a good Pauling bond order-coupling constant correlation will exist. The ortho proton coupling constants for 50 sets of positions in 14 polynuclear aromatic h y d r o c a r b ~ n s l * . 'fit ~ -eq ~ ~4 with a correlation coefficient of 0,984, and an average deviation of fO.1 Hz. In exact corre-
5.27 + 4 . 3 4 9 + 0.54s (4) spondence to the former work with Coulson bond orders, one finds that it is necessary to include a term for enhanced magnitude of a coupling constant if one of the coupled protons is in an overcrowded environment.'>.18.j7,sxThe term S in eq 4 refers to the 1,8 pair of protons in phenanthrene IV and is a correction to be applied to J H H ' , * The . statistically determined enhancement, 0.54 Hz, is larger than ca. 0.3 HI. found previously.58 No statistically meaningful correction is necessary for the 1,8 protons in naphthalene where a value of ca. 0.1 H z has been used.j8 JHHortho=
IV Theoretically, the enhancement effect is probably due to several factors. The buttressing effect of the overcrowded protons which shortens the coupled proton-proton distance and the resulting decreases in H C C angles would lead to increased coupling constant, whereas out-of-plane distortion of the molecular framework would give a d i m i n ~ t i o n . ' ~ , ~ ~ The resulting change in coupling constant does seem remarkably constant from molecule to molecule considering the widely varying geometries of aromatic hydrocarbons (see Table 111 for references). One concludes that the various kinds of distortion from ideal geometry accidently cancel to a large degree in their net effect on ortho coupling constants. 59 An interesting application of eq 4 is to use nmr data to calculate bond orders in compounds with unusual structural features for comparison with theory. Agreement should be no more than semiquantitative considering the sensitivity of coupling constants to bond angle changes. When applied to
Herndon
/ M O Formulations of Pauling Bond Orders
7608 Table 111, Bond Lengths in Benzenoid Hydrocarbons (A) Compd
Bond
Bond order
Naphthalene (1961p
a
0.333 0.667 0.333 0.333
b C
d Anthracene (1964)b
a b C
d e Tetracene (1961)"
a b C
d e f g Pentacene (1961)d
a b C
d e
f g
h Phenanthrene (1971)~
a b C
d e f g h 1
Chrysene (1960)'
a b C
d
e f g
h 1
j k
Benzphenanthrene (1963)g
a b C
d e
f g
h 1
j
k
Triphenylene (1963, 1973)h
a b C
d e Pyrene (1972)
