William C. Herndon and Cyril Parkanyl Universltv 01 Texas at El Paso
I
a Bond Orders and Bond Lengths
The relationship between theoretical bond orders and bond leneths in unsaturated hydrocarbons provides a pedagogically n s e h comparison of results of qnant&chemicd calculations with experimental data. In most cases the hond orders have been calculated by the Huckel molecular orbital (HMO) method, or by one of the variants of the self-consistent-field (SCF) technique, and then hond lengths correlated with bond orders by a linear length-order equation (1,2). The agreement of predicted hond lengths in benzenoid hydrocarhons with experimental values is usually considered as a support for the concept of a hond order. A recently descrihed empirical strncture-resonance theory limited to Keknl6 structures (33) gives results for many properties of unsaturated hydrocarbo& that are in agreement with the results of SCF-MO calculations. In fact, the agreement is so precise that the resonance theory calculation can he taken as eauivalent to applying an SCF-MO procedure. However, the iesonance t h G y merhod has a significant nd\,antage in that calculations are rtrrrled out by hand in a fraction of the rime needed for the MO calculations. Bond orders from a resonance theory limited to Kekul6 srrurtures are ralled Pauline bond~ orders. and ~ the formulation . . - ~ ~ ~ ~ ~ - ~ of a Pauling hond order-bind length relation actually preceded the MO definition ( 6 ) .For particular molecules it had been pointed out that Pauling bond orders were as successful in exnlainine hond leneths as anv of the other theoretical methbds (7->). A recent more complete compilation (5) of hond orders and bond lengths shows that the Pauling method is generally successful in correlating the bond lengths of olefins, henzenoid, and nonbenzenoid cyclic hydrocarhons. In the present paper, we will compare the actual correlative abilities of the HMO, SCF-MO, and the Pauling theories by statistical techniques. Previous discussions havebeen limited to aualitative statements regarding accuracy, or a t most the average deviation of n few eiperim~ntaland predicted hond lennths has been ohtained. We will also delineate several additional points that we believe should be discussed in a presentation of the bond order-bond length relation to students. ~
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Deflnltions of Bond Order In this section the well-known methods for calculating bond orders will be summarized, and some alternative lesser known ways will be outlined. In addition some mathematical equalities between the results of MO calculations and the Pauling bond orders will be given. I t should he mentioned that a unique definition of hond orders is not obtainable. The hond order should somehow measure the electron density between two nuclei, but the exact manner in which the electron density is partitioned in various regions of amolecule can he a matter of choice. MO Calculations
Huckel molecular orbitals are defined as eigenfunctions of an effective one-electron a Hamiltonian within a nearestneighbor-only interaction (tight-binding) approximation neglecting overlap. The Huckel MO's are identical to those Presented at the Second International Congress of Quantum Chemistry, New Orleans, Louisiana,April 19-24, 1976.
obtained by solving the topological matrix (I0,II) or the adjacency matrix of the graph of the molecular system (12). HuckelMO's are physically meaningful because short-range forces are much larger than long-rangeforces in molecules, and the tight-binding approximation expresses short-range character. The original hond order definition given by Coulson in terms of Huckel MO's (13) is shown in eqn. (1).The sum is over j occupied levels
PF. = xi n,cj,cjz
(1)
n, represents the number of electrons in level j , and the coefficients of bonded atoms r and s are c,, and cj,. The Coulson bond order is an inferential measure of the electronic charge in a particular bonding region but it is not an electron density or bond population. The sum of all electron densities or populations is the total number of electrons, whereas the sum of all Coulson hond orders gives the total a energy of a system (14) including the delocalization energy, eqn. (2). E , = ~ X P C I ~
(2)
The Coulson bond order is therefore related to binding energy contributed by a bond, and the more positive is the bond order, the larger is the force binding two nuclei. Overlan is included in manv s i m ~ l ea~olicationsof MO theory. ~ i p i c aresults l can be f&nd for n s?;cems in the tahles of Streirwieser and Coulson c l ; j r . or in the Daoers bv Hoffmann on the all-valence-electron e x t e n d e d ~ i c k emethod l (16,17). Mulliken proposed the function given in eqn. (3) to represent bond order under these conditions (18). p: = (1 + S ) (n,cjrcj,) + (1+ cjs) (3)
+
The leading factor (1 S ) is added so that a hond order of unity will be obtained for ethylene in its ground state indicating one full a bond. The Mulliken bond orders are called overlap populations, and the same comments regarding the physical meaning of the Coulson bond order also apply to the Mulliken hond order. There is an essentially linear relationship between Mulliken and Coulson hond orders for a systems where the overlap integral is given a constant value (S = 0.25). The derivation of this relation is given in naners . bv Ham and Ruedenberg (19,201. Self-consistent-field (SCF) MO calculations restricted to a electrons (pariser-par;-~o&e (21)) and of the all-valenceelectron tvne (22). . . . neglect the .. (CNDOW . . .. MIND0 (23)) overlap integral in order to avoid problems in calculating inteerals involving electron reoulsion. Conseanentlv bond orders are calculated according tokqn. (i),and the interpretation of an SCF bond order is vew similar to that of the Coulson HMO bond order.
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Pauling Bond Orders
The least efficient way to determine Pauling hond orders is to draw all Kekul6 structures of the molecular system, and determine by inspection the ratio of structures in which a particular hond is double to the total number of structures, eqn. (4). It is true that a particular advantage of Pauling = NDISC
(4)
bond orders is that the bond orders can be determined from simple drawings of the molecular graph. However, in most cases, one or two drawings will suffice, and a laborious count Volume 53, Number 11, November 1976 / 689
of the single and double bond rhararter of individual honda is not necessary. Outlines of many different methods for ohtaining bond orders and structure counts (SC) for unsaturated molecules are given in some recent articles (5,24). The most simple procedure for determining Pauling bond orders has been proposed by RandiE (25). His procedure al! from examination of lows a determination of SC and . D smaller subgraphs of the original iolecular graph. To obtain the SC, any arbitrarily chosen bond is fixed to be first double, and then single. The sum of the SC for the residual graphs remaining after deletion of all remaining essential single and double bonds is the SC for the original molecule. The bond order for the initially chosen bond is therefore known, and the bond orders of the adjacent bonds follow, since the sumof the Pauling bond orders of intersecting bonds must eaual unitv. The prkedure is illustrated for some of the bonds of be;zanthracene in (I).
Brockway, and Beach (6). A theoretical rationale for such empirical relationships bas been outlined by Salem (21, which simply depends upon the fact tbat the delocalization energy in HMO theory is a function of the bond order and the resonance integral p, eqn. (2), and p in turn is a function of bond length. The m bond energy is therefore a function of bond length as is the rr bond energy. For a hond a t its equilibrium length, eqn. (6) can be written
d',+d',=o br
(6)
and because of them energy, bond order, 0, and bond-length interrelationships, bond order must be a unique function of r. The relationship is not necessarily linear, and Coulson (13) derived eqn. (7) r = s - (S - d) + (1
+ [hslkd][(l- p)lp])
(7)
where r is the length of the bond to be calculated, s and d are the lendhs of Dure sinale and double bonds.. resoectivelv. . .. the k's areyhe res&tive 6rce constants, and p is the theoretical \10 bond urder. However. ~ n ~ h a bthe l v most used 410 bond order-bondlength relations are linear equations, as in eqn. (8)
(1) Of course, if the residual araphs are verv large their SC can be determined in the samem.anner. 1f one ob&s a subgraph for which no classical Kekul6 structure is possible, then the original deleted bond has a bond order of zero. This is illustrated for the central hond of zethrene in (11).
(11)
The residual structures are radicals for which no classical Kekul6 structure is possible. One of the most interesting aspects of the Pauling bond order concept is the fact that the Pauling results can be derived exactly from the eigenvalues and eigenvectors obtained from HMO calculations. If one defines an MO bond order similar to the Coulson bond order, eqn. (11, but weighted by the inverse of the eigenvalues, eqn. (5) is obtained P,S = X njc,c,lq (5)
In this definition, higher occu~iedenergy levels contribute more to the bond ord& than rhk lower lev.ek as is found in the Mulliken bond orders, eqn. ,374. Hart was the first w reropize that rhe bond orders given by eqn. (5) are precisely identiral tu the I'auling wlues (26).The reahon for the identity has been eluridated h\, Ham. who showed that it onlv holds for even alternant compounds (27). Equation (5) shows tbat one can use tables of HMO eigenvalues and eigenvectors to determine Pauling bond orders. More imnortantlv. it demonstrates an aspect of the essential topological character of both the HMO and Pauling concepts of bond order. They are related concepts but there cannot be any exact linear correspondence hetween the two calculated orders. cf. ems. (1)and (5). In addition. the Pauline bond order has a chara&ristic of a bond density in that thesum of all Pauling hond orders for a molecule is the total number of m electrons. The Pauling bond order bears a resemblance to the Coulson bond order, but is analogous to a bond population. Bond Orders and Bond Lengths Empirical comparisons of bond orders with bond lengths can be found in the earliest paper on resonance by Pauling, 690 / Journal of Chemical Education
r=a+bp (8) pruposed, for example, by Coulson and Golehiewski (2d1 w d Pauling bond orders have also been 1)ewar and de I.lano t 2 9 ~ assumed (51to havea linear correspondence with hond lenmh. The various empirical relationshi~shave been reviewed hv Trotter (30). Experimental Bond Lengths Usually a few standard molecules whose bond lengths are known very accurately are chosen to determine the constants in eon. (8). Presumahlv, the statistical analvsis of all known bond lengths and bondorders would yield the best correlation of data, but the ~rocedureof usinn a small number of optimum hond lengths has a useful advantage. If the correlation for the standard molecules is precise, then one can consider large deviations from the theory as special cases for which ad hoc explanations may be necessary. Structural effects might he discussed in this way, whose identification might be obscured by the statistical fit of all known values. Also, it is important to realize that X-ray diffraction bond lengths may not be as accurate as quoted experimental errors would indicate (31). For example, an examination of the available neutron and X-ray diffraction data (32) suggests that bond lengths in bonds of high order are underestimated by as much as 0.02 A by X-ray diffraction (5).This is due to the fact that neutrons are scattered by atomic nuclei, whereas with X-rays the centers of electronic charge clouds are determined. Using the data from references (32) . . one finds the roughly linear relationship given in eqn. (9) Ar(Neutron - X-ray) = 0.032pP - 0.007 (9) An additional last factor is concerned with the age of experimental data. Many of the supposed anomalies in bond order-bond leneth relationshins have vanished when more accurate experimental data have become available. The central formal sinnle bond in butadiene is a eood example of a bond which hay gradually shortened u,ithL~imc from.l.483 h r . U ) to theoptimum v a l u ~ o1.4fi5 t A (3474 in betteraereement with nearly Hi1 theoretical calculations. kdditional examples are the short bond in pyrene, from 1.320 A (35)to 1.367 A (32), and the interring bond in perylene from 1.50 A (36) to 1.471 A (371, c.f. {III),also leading to better correlation of theory and experiment.
@ 8 cm
Table 1.
Exoerimental ( A ) and Calculated Bond Lengths
Compound
rlexprl
Benzene Ethylene Butadiene Butadiene Graphite
1.397~
1.3376 1.343C 1.46Sc 1.421d
pC
0.667 1.000
0.894 0.447 0.535
~Icaicdl
1.402 1.327 1.351
0.500
1.452
0.000
1.432
0.333
aye. dev.
1.401 1.339
1.000 1.000
0.979 f0.009
C O ~ coeff. .
Table 2.
Pp
rlcal~dl
1.339 1.463 1.421
0.998 f
0.002
rlcalcd
- SCFI
1.39@
1.398f
1.308 1.344 1.330 1.468 1.464 1.419 0.999 1.338
f0.002
Table 3. Pyrsnopyrene and Corannulene, Bond Orders and Band Lenathr
Bond Length Correlation@ Altsrnanf Comooundr
Ethylene Butadiene Hexatriene Benzene Naphthalene Anthracene Tetracene Pentacene Number Of bonds Corr. coeff.pC AYB. deviation Corr. coeff.pP AVO.deviation Corr. coeff. SCF Ave. deviation
Pyrenopyrene a b C
d C
f 9 h i
1
K avtt~agedeviation Corannulene
1.402 0.727 1.391 0.455 1.413 0.273 1.440 0.273
b
QSee text for references.
d
average deviation
There do not seem to be any exceptions to this trend of better agreements with calculations, the more recent the experimental data. Correlations of Bond Length Data
The optimum experimental values of hond lengths in benzene, ethylene, butadiene, and graphite listed by Allinger and Sprague (34) are given in Table 1.Also listed are Pauling and Coulson bond orders. calculated bond lenaths from least-squares regression analyses, and bond lengths&hlated by SCF-MO methods. The last column of bond lengths was calculated by the latest version of the all-valence-electronSCF procedure (MIND013) developed by Dewar and his coworkers (38). The relatively poor agreement with experiment is simply a reflection of the fact that MIND013 is parameterized to primarily fit heats of atomization for a large variety of molecular types (39), rather than just to fit molecular geometries of hydrocarbons. For the latter purpose, the SCF method of Dewar and de Llano (29) is highly satisfactory. For the small group of molecules in Table 1, the Pauling bond order linear eqn. (10) is just as accurate as the more elaborate SCF calculations. r ( P ) = 1.463 - 0.124 pP
(10)
The relatively poorer agreement of a linear Coulson bondorder equation, given in eqn. (11) r ( C ) = 1.552 - 0.225 p C
(11)
is not surprising. The difficulty of fitting hutadiene data t o a curve containing graphite, benzene, and ethylene data has long been apparent (I). A least squares regression according to eqn. (7) improves the average deviation to *0.005 A, still less accurate than the results from the Pauling bond order curve. The 16 open-chain olefins and benzenoid aromatics given in Table 2, under the heading Alternant Compounds, contain 913 disrineuishable bonds. The structures and Pauline bond " orders are given in our earlier paper (5). HMO bond orders were obtained from standard tables (15) or by new calculations, and the SCF bond lengths were taken from an article ~~
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acaiculated according t o e m . (10). bcalculated according to e m . (11). csee reterense (43).
by Lo and Whitehead (40). T o conserve space the actual data are not listed. The errors and correlation coefficients for the alternant molecules were obtained by unweighted least sqaares regression analyses that yielded the linear eqns. (12), (13), and (14). r ( P ) = 1.465 - 0.130pP
(12)
r ( C ) = 1.573 - 0.273 p C
(13)
r = -0.060
+ 1.04 r(SCF)
(14)
The correlation coefficients are practically identical, the calculated value of 0.92 indicatine that annroximatelv 85% of .. the variation in bond lengths is accounted for by thevariation in bond order. The averaee deviations of calculated and experimental values are very similar from the three different techniques, and they compare well with the estimated experimental errors of *0.008 A. Structures, SCF and Pauling bond lengths, and references for the non-alternant systems can also be found in Ref. (5). The hond lengths using bond orders were calculated from eqns. (12) and (13), and SCF bond lengths were taken from various literature sources. The average deviations are approximate1 the same as the estimated experimental errors of 3~0.015 It should he observed that the experimental structural data are actuallv for methvl or nolvmethvl derivatives except in the case ojazulene. +he effefect of the methyl groups on internal bond lengths is not expected to be important. Our conclusions are that Coulson HMO bond orders, Pauling bond orders, and LCAO-SCF-MO bond orders are eauallv . effective in ~redictinaand correlating bond lengths in a-systems. One therefore expects the different theories to correlate well with each other, and this is correct. The correlation coefficients are SCFIHMO, 0.950, and SCFIPauling, 0.985. The correlative ability argues for the essential correctness of the bond-order concept, determined by any of the
1.
Volume 53, Number 11, November 1976 / 691
theories. T o us, the simplicity of the theory and the ease of calculations using Pauling bond orders make it the method of choice for an application of this type. We even postulate that -laree Pauling bond lengths ~ - deviations from the ~redicted are probably the results of experimental error,-and that more accurate experimental results will resolve these discrepancies.
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A Bond Order-Bond Length "Experiment"
Recent accurate X-ray data are available for alternant pyrenopyrene (IV) (41) and nonalternant corannulene (V) (42).
The prediction of hond lengths in these molecules using only the data for ethylene, butadiene, graphite, and benzene provides a good demonstration of the hond order-bond length conceot. The student should derive his own bond orders and obtain bond order-length equations from the experimental bond lenethr riven in Table 1. A least squares analysis is not necessary if a careful graph is constructed. A summary of results is presented in Table 3. The average errors are approximately 10-20% of the total range of hond lengths, as would be predicted from the correlation coefficients given in Table 2. The relatively poor agreement of predicted Coulson bond lengths for pyrenopyrene can he corrected to f0.013 A if eqn. (13),based on all bond lengths for compounds in Table 2, is used. final ~ o i nof t interest is the almost exact correspondence -A- -----of the calculated SCF and Pauling hond lengths incorannulene. This correspondence is provocative but probably coincidental since the SCF values were obtained (43) for anonplanar dish-shaped geometry of the molecule. ~
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Acknowledgment
We gratefully acknowledge the financial support of the Robert A. Welch Foundation.
692 / Journal of Chemical Education
(1) S t r e i t ~ e ~ eA., r , J r , "MoleeuLu Orbital Theory for Organic Chemists..) John Wileyand Sons, NeuYork, 1961,pp. 165-172. (21 Salem. L.. 'Tho Molecular Orbital TheoryofConjugsted Systsms.)'W. A. Benjamin. NeulYork, 1966.p~.13C148. I31 Herndon, W. C., J Amar Chem Soc., 95,2404 (19731. (4) Herndon. W.C.,and Elhey, M. L.,Jr.,ilAmar Chem S o t , 96.6631 (1974). (61 Herndon, W.C., J. Amar. Chrm Soc.. 96,7605 (1974). (6) Paulinp, L., Brackw&, L. 0.. and Beach. J. Y.. J . Amer Chem. S o r , 57, 2705 11935). (7) Cruiekshsnk, D. W. J., and Sparks,R.A.,P~oc.Roy.Soc.. Ssr. A, 258,270 (19601. (31 Cnickshsnk.D. W. J., TeIioh~dion.17.155 (19621. (9) Couhon. C. A . in "Physical Chemistry, An Advanced Treatise," Val. 5, Academic Prerr, New York, 1970, p. 381. (101 Ruedenberg,K., J. Chem. Phys, 22.1876i1954). ~ . Ruedenberp, K . , J Chem. Phvs., 29,1199l19581. (11) H ~N. %and (12) Gutman. I., and Trinajsfit, N.. Topics in Current Chemistry, 42.49 0973). (13) Coulson.C.A..Proc.Roy. Soc.. SPI.A, 189.413 (19391. (1" C~ou1son.C.A.,and Longuet-Higgins, H. C.,Proc. Roy. Soc.,Ser A, 191.39 (1947). (151 Cou1son. C. A., and Sfreifwiener. A.,Jr.."Dictionaryof r-Electron Calculationr," W. H. Fremsn, Sen Fmncisco. 1965. (161 Hoffmann,R., J. Chem. Phys.. 39,1397 (19631. (171 Review article: Herndan. W. C., Pmg. Phy8 Org. Cham.. 9.99 (19721. (181 Mulliken, R. S . , J Chem. Phys.. 23,1633,1841 (19551. (19) Ham.N.S..andRuedenberg, K.. J. Chsm. Phys., 29.1215 119581. 1201 Ruedenberg. K., J . Chsm. Phyr., 34,1684 l19611. (211 Pople, J . A,, Trans. Faraday Soc., 49,1375 (1953): Psriper, %and Pam,R. G., J. Chsm. Phys., 21,466,767 (1953). (221 Pople,d.A..Santry,D. P.,and Sees1,G.A.. J. Chem Phyr., Suppi., 43.512911965): Pop1e.J. A,, and Segel,G.A., J
[email protected].,43,5136(1965):44,3289 (19661. 1231 Baird. N. C.. and Dewar. M. J . S., J. Chem. Phys., 50,1262 (1969):Dewar. M. J. S.,snd Hasolhsch,E., J. Amm Chem. Soc., 92,590 119701: Dewar,M.J. S.,andLo,D.H.. J. Amer Chem Soc.. 94,5296(19721. 1241 Herndon. W. C.. T~trahadron.29.3 (1973):J. CHEM. EOUC..51.10 (1974). 1261 Randie. M.. Crooiic. Chem. Alto. accepted for publication. 126) Plstt. J. R..quotad inreferenee (191. 127) Ham.N.S.. J Chem Phys., 29.1229 (19581. I281 Golqbiewski. A.,Pmc. Phys Soe ILandoni, 78.1310 (1961). (29) Dearar, M. J. %,and deLlano,C.,il Amar. Chem.Sor., 91,769(19691. 1801 Trotter. J.,Roy. Insf. Chem Lecl. Ser. 2,1(19641. 1:IlJ Coppons. P., Arlo Cryst., 830,255 119741. (32) Phenanthrene, Kay, M. I.. Okaya, Y., and Cox, D. E.,Aclo Cryst., B2G 26 (1971): Triphenylene, Ahmed, F. R.. and Trotter. J., Acto Cr)'rt., 16, 503 (19631, Ferraris, G . , dones, D. W.. and Yerkers, J.. 2. Kristoilagr. K r i s t d k o a m ~ t r iKrinailoh~s., ~, K~irtollchem.138, 113 119731;Pyrene, Hszell, A. C., Larsen, F. K., and Lehmen, M.S..Acta Crysf.. B28, 2977 119721. o~ ~ , s m d . , 12,1221 (88) A I ~ ~A,, ~~a s t~i s n a~~0.. n ,~andz~ ~~ ~ ~ t ,tM.,e ~b c et them. llii
$ 1 1 np.r, h I.. aud 'iprsw..l. T.J Amrr Cnsm S o < . 95. Idlc 15-:, i ' m . r m n n ..snd'l'rrr.rr 1 . 4imCr:d ln.d!6 136: ,C 1 ) .A j. m D \I Rohcrson J \I .an
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