Molecular orbital calculations of bond lengths in benzenoid hydrocarbons

Te4sddmn.

191%. Vol. 22. pp. 3381 to 3392.

MOLECULAR LENGTHS

Rqamon

Ra

LtA.

PrimteA In Nonbcm

IdanA

ORBITAL CALCULATIONS OF BOND IN BENZENOID HYDROCARBONS

G. v. BOYD Department of Chemistry, Chelsea College of Science and Technology, London S.W.3

and N. SINGER Department of Chemistry, Northern Polytechnic, London, N.7 (Received 24 May 1966)

Abatraet-The correlation between bond lengths and bond orders in benzenoid hydrocarbons has been considered. Bond orders for six molecules were obtained by means of a simple MO-LCAO-SC treatment, and a procedure is suggestedfor calculating accurate bond lengths from such selfconsistent bond orders. ONE of the conspicuous successes of the Htlckel molecular orbital (HMO) method is the prediction of bond distances in conjugated hydrocarbons. An excellent summary of recent work in this field has been provided by Trotter.’ In this paper we use a refined version of the Hiickel method for bond order calculations and examine its application to the prediction of bond lengths. We confIne the discussion to planar benzenoid hydrocarbons for which bond distances are known with some accuracy. These hydrocarbons are altemant, a circumstance that simplifies the treatment. A bond length calculation involves two separate steps: (a) the calculation of bond orders, and (b) the derivation of bond lengths from a suitable bond length-bond order relationship. We consider these steps in turn. Calculation of bond orders

For a neutral conjugated hydrocarbon r-s bond is given by

with n carbon atoms, the bond order of the n/2

PC. = 2 F,

CkfL

(1)

where k enumerates the molecular orbital, and the co-efficients c, c,, are the elements of the k-th eigenvector of the Hamiltonian matrix, whose elements are given by

H r, =

$,*H+, dr = B,,, r # s (3) I In the simple HMO theory it is assumed that all Coulomb integrals, cc,, are equal (= a), and alI resonance integrals, /?,,, are equal (= B) for bonded carbon atoms and zero otherwise. The assumption that all Coulomb integrals are equal is just&d for altemant hydrocarbons by Co&on and Rushbrooke’s theorem.% However, equating all resonance integrals to /I leads to an inconsistency since the resulting bond orders, 1J. Trotter, Crystal-Structure Studies of Aromutfc Hydrocarbons, Royal Institute of Chemistry Lecture Series, 1964, No. 2. * C. A. Co&on and G. S. Rushbrooke, Proc. C’ambrfd’gePM. Sot. 36,193 (1940). 3383

3384

G. V. Bon, and N. SINGER

which are not equal except in monocyclic hydrocarbons, show that the n-electrons are not uniformly distributed over all the ca.rbon+arbon bonds. The Htickel Hamiltonian matrix is thus self-consistent with regard to its diagonal elements but not with regard to its off-diagonal elements. We would expect molecular orbital calculations to yield better results if the matrix were fully self-consistent. This can be done by making the resonance integrals depend on the bond orders:

where B is the benzene resonance integral, the exact value of which is not required. The Hiickel bond orders thus give rise to a new set of resonance integrals, and the new matrix is again solved and a third set of resonance integrals is calculated, and the process is continued until the bond orders no longer change. The Coulson-Rushbrooke theorem still applies in this situation, so that the Coulomb integrals are not affected. The Hamiltonian matrix is now fully self-consistent and the new bond orders are an improvement over those calculated by the HMO method. The exact form of the function (4) is of importance Lennard-Jones3 suggested a parabolic expression, but in more recent work Longuet-Higgins and Salem4 preferred an exponential relationship as the resonance integral must decrease rapidly with increasing bond distance and hence with decreasing bond order. This question is discussed further below. Using an exponential function, Coulson and Golebiewski5 calculated the bond distances in naphthalene and anthracene with very encouraging results. These authors obtained an approximately self-consistent Htickel matrix by an elegant perturbation procedure rather than by the iterative process outline above. Calculation of bond lengths Having obtained a set of self-consistent bond orders, pld, we must now derive the corresponding bond lengths, R,,. It is generally assumed that the length of a bond linking trigonally hybridized atoms depends solely on its n-bond order. Two relationships have been used: Coulso# derived

where s is the “natural” single bond distance between sp2 hybridized carbon atoms, d is the double bond distance in ethylene, and k is a parameter. In later work, a linear relationship of the form

assumed. The parameters b and c can be obtained from the bond lengths and bond orders of ethylene and benzene and will clearly depend on the exact values of these

is

’ J. E. Lennard-Jones, Proc. Roy. Sot. A1!58,280 (1937). ’ H. C. Longuet-Higgins and L. Salem, Proc. Roy. Sot., A251, 172 (1959). 6 C. A. Coulson and A. Golebiewski, Proc. Phys. Sot. 78, 1310 (1961). a C. A. Coulson, Proc. Roy. Sot. A169.413 (1939).

Molecular orbital calculations of bond lengths in benzenoid hydrocarbons

3385

quantities. Coulson and Golebiewsk? used 1.337 A and 1.000 for ethylene and 1.397 A and 0667 for benzene and obtained KS = 1.517 - 0*18Op,, This differs appreciably Higgins and Salem4

from the corresponding

equation

(7) obtained by Longuet-

&, = 1.50 - O*lSp,,

(8)

using a different vafue for the bond fength of ethylene. The resonance integral-bond order relationship

In order to obtain the relationship of E!q. (4), Longuet-Higgins suggested that dependence of 8 on R be written in the form Jj = -Be-JV@

and Salem4 (9)

By considering the theoretical expressions for the force constants for the totally symmetric and totally antisymmetric vibrations of the benzene molecule together with Eq. (6) they showed that the ratio of these two force constants, X, was given by

An examination of analyses of the vibrational spectrum of benzene lead to a value of l-9340 for X so that a could calculated from the known value of c. Using c = O-180 leads to u = Q3727. These values give the equation & = /I exp (O-48294pr8 - O-32196)

(11)

where ,!Iis the benzene bond integral. This is the equation used by Coulson and Golebiewski. At first glance this analysis appears to imply that the numerical values in (11) depend on the bond lengths and bond orders of the bonds in ethylene and benzene since these are required to find the numerical value of c and on the ratio of the force constants of benzene X; a total of four experimental parameters. Further, any change in the parameters of (6) should lead to a new value of a and thus to new values for the parameters in (11). A slightly different analysis shows that in fact the parameters in Eq. (11) depend only on the value of X and on the bond order in benzene. This means that any linear relation of the form (6) will be consistent with (11) for a given value of X. We start our analysis with Eqs (6) and (9) from which it follows that

(12) where R = benzene bond length and R=b-$c

(13)

3386

G. V. Bon> and N. S~OER

where we have introduced the bond order of benzene = &. Combining equations (6), (12) and (13) yields $=exp

(i(p_-g))

(14)

From (10) c X-l -=-

a

x

so that only one parameter establishes Eq. (11). This fact allows the possibility of choosing the parameters in Eq. (6) in different ways whilst at the same time retaining self-consistency with Eq. (11). Also, since the value ofp, for benzene must be 4 in our MO theory, we see that X is the only experimental parameter which determines the constants in (11). It is, therefore, possible to examine the agreement between calculated and observed bond lengths as a function of X. The most general formulation of Eq. (11) would thus be B,, = B exp Np,, - O-667) (15)

where h is a constant depending only on X. Numerical values

In testing whether a bond length calculation using a self-consistent Hiickel matrix is an improvement over the simple HMO method we are restricted to hydrocarbons whose bond distances are known very accurately. Only naphthalent and anthracene meet these requirements. Because of the differences in the results of bond length determinations by different methods it is not always easy to decide which set of experimental results to use for purposes of comparison with the calculated values. We therefore compare our results with those from X-ray and electron diffraction and with the average of these two methods. Table 1 compares the results of our SC calculations using Eqs (7) and (11) with the experimental results. For comparison HMO bond length obtained by means of Eq. (7) from HMO bond orders and from SC bond orders using Eq. (17) as well as VB lengths are included. Coulson and Golebiewski’s results6 are not quoted as they are almost identical with our own SC results. All the calculations reported here were done on the unmodified Htlckel matrices of the molecules using the London University “Atlas” computer. Iterations were carried out until bond orders in successive cycles differed by less than 0.001; 4-5 iterations were usually sufficient to achieve convergence. We use C 6 as a measure of the deviation of the calculated from the experimental results. This is obtained by summing the absolute value, multiplied by 108, of the difference between observed and calculated bond length. Alternatively, & = (Z 6)/n, where n is the number of bonds considered, gives the mean deviation per bond from the experimental value. The results show that the iterative method and the Coulson and Golebiewski pertubation methods lead to virtually identical bond lengths. These self-consistent bond lengths are a considerable improvement on those calculated by the HMO method, particularly when the average of the X-ray and electron diffraction results are compared with the calculated values. It is worth noting that the results

Molecular orbital calculations

Naphthalcne

of bond lengths in bcnzenoid

hydrocarbons

3387

Anthracene

Pcrylene

chryeene FIG.

Pyrcne

1. Numbering of bonds in the molecules.

given by the simple Pauling superposition method are slightly better than the simple HMO results if the X-ray data are used in the comparison but considerably worse if the electron diffraction data or the average of these and the X-ray data are used. Further, the Pauling method does not reproduce the variation of length from bond to bond which the simple HMO method gives and which agrees in direction if not in value with the experimental results. The question now arises whether further improvements in calculating bond lengths are possible within the framework of an iterative HMO scheme. An obvious way is to retain the linear bond length-bond order relation (6) but to use values for the parameters b and c which differ from those in E$ (7). As shown previously such a procedure is completely consistent with Eq. (11). Such a course was adopted by Cruickshank and Sparks’ who drew the “best” straight line through 34 points on an observed bond distance-HMO bond order diagram, and found R_ = l-567 - 0.267~~.

(16)

This equation gives a great improvement in the calculated bond lengths for benzenoid hydrocarbons,’ based on HMO bond orders. We have carried out a similar procedure to obtain the parameters b and c in Eq. (6) but instead of using the HMO bond orders a self-consistent set of p,, values was used to draw the “best” straight line. The calculations were extended by not only using Eq. (11) but also (15) with a series of diEerent values of h in order to determine whether a change in this parameter could improve the calculated bond lengths. We choose a plausible value for h and do a self-consistent bond order calculation for naphthalene ’ D. W. J. Cruickshank and R. A. Sparks, Proc. Roy. Sot. A258.270 (1960).

TABLE

t.

~~AR~~

OF

OBSERVED

AND

CALCULATED

BOND

LENGTH

Expcrimcntal SC Method I4 A

C6 6%

(7)

WMOC

VW

SC Method 11’ _--~-.

c

X-Ray’

Elec. &If.“’

IX f B/Z

Rw

t: c

1‘364 1.421 I*415

1,371 1.422 1.412

1,368 1422 1.414

1.382 1.420 I+4t4

d

1.41%

I.420

1419

I-419

a b c d e

1‘368 i-436 1.399 1,419 X.428

l-393 1.420 I-4@4 1.419 I.425

1.379 t-428 I*402 I.419

I.378 l-427 1406 I*420 1,428

Go&

GPM

B

Ea.

Calculated

Malede

and bond

USN%

1,427

&dp 8, 18 -1 -f I

sc

8,

6s

sc

8,

II * 14 -2 -2 2 0

23 -4 -6

16 -5 -3

19 -5 -5

7 1 7

-I

0

10 -12 --I -9 7 -1 7 2 4 1 1 1 48 0 41 3 24 1 5

5

a Self-consistent bond orders calcuiated by means of Eq. (11). This paper. b S, is the deviation of the calculated bond length from the X-ray data x ICT etc, c Only 6 values are quoted. d Ref. f . Only S values arc quoted. * SC bond orders from Eq.(17). This paper.

3

645

0 0 fa

se 3 0 8

423

16 -6 5 -15 1 -7 9 4 6 -7 --I -I 8% 2 51 5 62 3 10

6,

6

7

88

17 0 I

10 13 -1 -1 4 2

c, 4 ;a

-2

3

0

-7

-29 -I8 3 I9 IS -f -10 -8 20 20 20 65 11 la4 14 83 12 7

6,

12

9

sc

-I

8 -14 -3 -8 8 0 6 1 3 3 3 3 441 45 2 26 0 5

s

3

g z ;, 8 51:

Molecular orbital calculations

of

bond lengths in benzenoid hydrocarbons

3389

and anthracene (nine different bonds), using Eq. (15). We next determine the regression line, i.e. Eq. (6) for the bond length-bond order correlation for these 9 points together with the benzene value and calculate the sum of the absolute deviations of differences between calculated and observed bond lengths C 6.* The calculations are repeated for a different value of h and so on until the smallest x 6 value is obtained. The corresponding values of h, 6, and c are the best parameters and can be used in bond distance calculations for further planar conjugated molecules. A series of six such calculations were carried out for values of h ranging from O-3 to 0.8, the regression lines being based on X-ray diffraction results. A further series of calculations was done with h varying between 0.4 and 0.6, this time basing the regression lines on the electron diffraction data and on average of the X-ray and the electron diffraction data. The results are summarized in Tables 2 and 3. The bond length of benzene was assumed to be 1.397 for all these calculations. An examination of these results shows that variation of h between O-4 and 0.6 does not affect the fit of the calculated results very greatly. However, the X-ray data can be fitted best by an h value of about 0.4, the electron diffraction data by a value close to 0.6 whilst the average of these two is best fitted by a value of h of 0.55. Finally we have used Eq. (7) to calculate bond lengths from bond order data obtained by using h = O-55and these results are included in the last column of Table 1. The results obtained by using the self-consistent method with h = 0.4829 and h = 0.55 and Eq. (7) give deviations from the mean of the X-ray and electron diffraction data which are almost identical and there is little to choose between these two methods. The “best line” method using these two values of h give results which again do not differ appreciably from each other or the Eq. (7) results although the largest deviations are somwhat reduced, as might be expected. There is little to choose between these methods since they all yield bond lengths to comparable accuracy for naphthalene and anthracene. We believe, however, that the substitution of h = 0.55 for 0.4829 in Eq. (11) together with the use of a “best line” based on the average of the X-ray and electron diffraction bond lengths leads to a marginal improvement of the calculated bond lengths since there is a reduction in the extent of large deviations from the observed values. Bond lengths calculated from self-consistent bond orders by means of the equations /?,, = @exp (0.55~~~ - O-3666)

(17)

and &a = 1.524 - O.l94p,,

(18)

should give bond lengths for aromatic hydrocarbons which are accurate to about 0.005 A. We have used these equations to calculate bond distances in a number of benzenoid hydrocarbons other than naphthalene and anthracene for which accurate carboncarbon distances are known from X-ray measurements. These bond lengths, while not known to such a degree of precision as those in naphthalene and anthracene, should be sufficiently exact (mostly to within OGO5A) to enable us to test our results. The results are given in Table 4 column (a). Also listed are the deviations of calculated * The quantity d was found to be a better indication deviation) emphasize the effect of a single large deviati0n.l

of the fit of the data since u (standard

3390

G. V. Bon, and N.

SINGER

TABLR 2. DEVIATIONS OF CALCULATED BOND LENOTIS BASED ON X-RAY AND ELECTRON DIFFRACTlON

h

DATA

AND

0.3

ON

THE

AVERAOE

0.4

OF THESE

FROM

0.4829

‘THE EU’ EIUMENTAL

0.55

VALUES

0.6

0.8

Molecule and bond W-h

CloH,

a*4

--6,

-1

-1

a b C

d a b

ClrHlo

8 0 -5

Cd &l C6 (Excluding G&4)

7 0 -3

6 -1 -8

4 -2 -7

44

C

d c

I%)

-2 7 48 55 41

0 6 40 33

dc,

0 -4 12 9 -4 -3 0 -2 -1 -1 -10 -7 3 -2 123 -2 -1 2 3 35 34 444 35 30

mA

6,

c

-1

0 -3 13 9 -4 -2 l-l -3 0 -11 -7 4 -1 123 -1 1 0 2 38 28 434 38 25

7 0 -2 2 -3 -6 3 4 37 30

-6 7 1 0 -1 -4 -5 5 2 34 28

B

0

m ).

0 -2 12 9 -3 -2 2 0 -5 -2 -11 -7 5 0 112 1 3 -1 0 41 26 534 41 24

-5 7 1 1 -2 -4 -4 6 0 32 27

B

1 13 -3 2 6 -11 -5 11 2 -2 45 53 44

C

4

-2

-3 9

9 2

-1 14 -4 -7 1

-11 -2 -2 1 22 -8 64 7 61

4 1 31 29

4 Sr gives the deviation of the bond length calculated from the best line based on X-ray data from the X-ray bond lengths in Table 1. *.OS, and do gives the deviations of calculated bond lengths based on electron diffraction data and on the average of X-ray and electron diffraction results from these results.

bond lengths obtained by the HMO method, using Cruickshank and Spark’s “best line” (Es. 15) (b), and those calculated by Pauling’s method (c),’ as well as the bond lengths from a “best line” calculation based on all the observed bond lengths and SC bond orders obtained with h = O-55(d). Overall the X b values for these four molecules are roughly the same by methods (a) and (b); 296, 291 respectively. Better agreement with the experimental values is obtained in (d) but the bond lengths of benzene, naphthalene and anthracene are not well reproduced. Since these are known more certainly than the other bond lengths we do not consider this method further. Again, the Pauling values are the worst and do not reproduce the change in bond length from bond to bond which is observed experimentally. 3. P AaAbiUERs

TABLE

AVERAQE

h

X-ray

b C

Ekditf.

b C

Av

b C

IN

&.

BOND

(6)

BASED

ON

X-RAY,

ELJXIRON

DIFFRACtION

LBNOTW FOR DIPFERENT VALUES OF

AND

h

@3

0.4

04829

0.55

0.6

0.8

1.5603 -0.2560

1.5559 -0.2487 1.5111 -0.1718 1.5334 -0.2105

1.5504 -0.2400 1.5067 -0.1645 1.5289 - 0.2024

1.5547 -0.2305 1.5021 -0*1570 I.5237 -0.1939

1.5397 -0.2223 1.4983 -0.1506 1.5193 -0.1865

1.5127 -0.1777

Molaxdar TABLE 4.

orbital calculations ci%fPARISON PmwLJmE.

OF

of bond lengths in benzcnoid hydrocarbons

CALCULATED

IRIPmNYLENE.

AND

OBSERVED

CIiRYsENE

Perylene’

(a) R.. a b C

d e f Triphenylene”

Chrysenels

is a b : e a b : e f g h i

PyrenelO

j k a b C

d F: Cd x& due (a) (b) (c) (d) (e)

I.370 1.418 1.397 1.425 1.424 1400 1.471 1447 1.415 I.416 1.377 1402 1.468 1409 1.381 1.394 1.363 1.428 1.421 1.369 1.428 1401 1409 1.380 1.420 1.417 1442 1.417 1.320

BOND

LENGTHS

FOR

PYRJZNB

Calculated

Obs. Compound and bond

AND

3391

1.381 1408 1.391 1.423 1.417 1.420 1a454 1448 1.410 1407 1.388 1403 1.435 1.414 1.382 1.410 1.381 1.416 1.430 1.371 1.427 1404 1.414 1.394 1405 1.419 1.436 1a425 1.365

d 11 -10 -6 -2 -7 20 -17 1 -5 -9 11 1 -33 5 1 16 18 -12 9 2 -1 3 5 14 -15 2 -6 9 45 296 218 8

Calculated by means of Eqs (17) and (18). Cruickshank and Sparks “best line” Ref. 7. VB Ref. 1. “Best line” based on all compounds (R,, = 1.5564 Neglecting bond a of chrysene and f of pyrcne.

(b) 6

(c) 6 8

-19 -2 1 2 20 -14 6 2 -10 6 5 -28 2 -3 8 14 -12 7 -3 -5 13 15 8 -12 10 -9 7 40 291 2.23 8

(d) b 5 3

-22 -4 -3 21 6 10 -10 -11 12 3 -34 6 0 21 18 -13 13 4 6 -4 6 17 -23 4 6 4 35 324 255 9

1 -12 -14 0 -7 21 -6 11 -7 -12 2 3 -27 4 9 14 8 -13 13 -12 2 -1 4 8 -18 3 0 1 29 262 206 8

0.2528 p,,) and equation (17).

Turning to an examination of the individual molecules, perylene presents some interesting features. The latest value for the bond gs is considerably less than the value of 1.50 previously found. This revision removes a great deal of the discrepancy between the observed and calculated bond length discussed by Co&on and Skancke.* Our ’ A. g C. lo A. I1 0. I* F. *’ D.

Camerman and J. Trotter, Proc. Roy. Sot. A279, 129 (1964). A. Co&on and P. N. Skancke, J. Chem. Sot. 1%2,2675. Camerman and J. Trotter, Acra Cryst. 18,636 (1965). Bastiansen and P. N. Skancke, Advances in Chemical Physics, 3, 323 (1961). R. Ahmed and J. Trotter, Acra C+. 16, 503 (1963). M. Burns and J. Iball, Proc. Roy. Sot. A257.491 (1960).

3392

G. V. Bon,

and N. SINGER

formulae give 6 for this bond 0.017 A and a bond length of 1.454 compared with the value of l-445 calculated by Coulson. Further the calculated lengths for the bonds b and c given by our formulae shows rather better agreement with the experimental results than the other calculations. Our calculations for triphenylene appear marginally better than those from the Cruickshank formula. (C 6 = 27 and 29 respectively). Again in detail we find that the b and c bonds are roughly equal, which agrees with experiment whilst the other calculations either differ more from the experimental results or suggest a bigger difference in the lengths of these bonds or both. In chrysene our calculations show no improvement over the other methods in predicting the length of bond a. The large observed value for this bond may be due to non-bonded H - * * H replusion.’ Overall, ignoring this bond we find &,r = 7-2 as compared with a value of 8.0 from the Cruickshank equation, again indicating a marginally better result from our procedure. The experimental results quoted for pyrene’O are probably not as accurate as the others. Most of the error is due to the bond f which is much shorter than the calculated values. Ignoring this bond, the 6, value of approximately 10 agrees well with the estimated u of the experimental results. It is difficult to decide if the difference between the observed and calculated length of this bond is a real effect due to some specific interaction neglected in our calculation or would be removed in a more refined structure determination. The estimated error in the length of this bond is given as u = 0.014 A so that a possible error of about 20 cannot be ruled out. As noted earlier a similar discrepancy between the calculated and observed length of the bond g of perylene was removed by a later structure determination. We conclude that on the available evidence our formulae should in general predict bond lengths of alternant hydrocarbons to about O-005 A, but will fail for very long or very short bonds or bonds in which some specific inter- or intra-molecular interaction occurs.