March, 1952
BONDLENGTHCALCULATION IN CONJUGATED MOLECULES
It may be emphasized that bonds corresponding to a completed sp3 octet for the second-row atom are to be distinguished from those involving orbitals beyond the octet. It is the octet bonds-the Lewis structures-that are of primary importance in determining the bond angles and general configuration of the complex, with the additional R bonds, involving d orbitals, effective mainly in changing the interatomic distances and the charge distribution. It might be possible to formulate an empirical system of quadratic energy terms, like those in Equation 6, that would reproduce many properties of molecules in a satisfactory may. Efforts ,to this
365
end have not yet been successful; even the simple treatment of Equation 6 fails when it is applied to molecules in which atoms of two or more kinds are attached to a central atom, and a similar treatment of the charge distribution (electric dipole moment) in molecules in which polar bonds interact with one another through induction has been found to be only partially satisfactory. The practical difficulties attending the quantum-mechanical treatment of the electronic structure of molecules are so great that the main hope for future progress in the treatment of molecular structure may, well lie in the development of a quantitative empirical theory of this sort.
BOND LENGTH CALCULATION IN CONJUGATED MOLECULES BY 0. CHALVETAND R. DAUDEL University of Paris, Paris, France Received January 6,196s
Hartmann has recently introduced the notion of bond type. The bond type in a purely conjugated hydrocarbon is determined by the number of the adjacent C-C bonds. Vroelant and Daudel, using the spin states method, have shown that to a rather good approximation Penney’s bond order for such a bond depends only on the types of the adjacent C-C bonds. A given bond is designated by a notation based upon the types of adjacent bonds. It is possible to show that there is for each notation a good relation between its exact Penney’s bond order and the Pauling double bond character taking account of only the Kekule formulas. I n this manner the evaluation of the Penney bond orders reguires only the calculation of the Pauling double bond characters which are very easy to obtain. Following an idea of Pauling and work by P. and R. Daudel, A. Laforgue has established an empirical self consistent method of evaluating the bond orders and the atomic charges in a conjugated molecule. This method uses the relations between interatomic distances, bond orders and the B integrals, and the relations between charges, electronegativities and the LY integrals. This method has been applied to the evaluation of t,he interatomic distances in some molecules like Ni04. The large N-N bond length found experimentally in this molecule is accounted for.
Introduction This paper is devoted to the description of some methods useful in the calculation of interatomic distances in conjugated molecules. In the first part we are concerned with a very simple method of estimating bond lengths in purely aromatic hydrocarbons. I n the second part we describe a possible way of evaluating with a rather good accuracy the bond lengths in some more complicated molecules.
A Very Rapid Method of Estimating Bond Lengths in Purely Aromatic Hydrocarbons The simplest method of estimating the bond lengths in aromatic hydrocarbons is to determine the “double bond character” using only the Kekule formulas as Pauling suggests.’ It is now well known,2 that the calculation of the bond orders using the molecular orbital method, or the valence bond treatment lead to better results. Evidently such a process is very laborious for large and non-symmetrical molecules, such as benzanthracene, for example. Here, we describe a method as simple as Pauling’s process but giving about the same accuracy as the molecular orbital treatment. It is applicable to the estimation of bond lengths in such polyaromatic (1) Pauling, “The Nature of the Chemical Bond,” Cornell Univ. Press, Itliaca, N. Y. (2) See for example Corilsou, 1)sudel sild Robertson, Proc. Rou. SOC. ( L o n d u n ) , i n the pmns.
compounds as naphthalene, aiithracene, pyrene, coronene, ovalene, and so on. Hartmann3 has recently introduced the notion of bond type. The bond type in a purely conjugated hydrocarbon is determined by the number of the adjacent C-C bonds. As an example the a-0 bond in naphthalene is designated as bond type 2 and the central bond as type 4. . Vroelant and Daude14 using the spin method, have shown that with a rather good approximation the bond order of such a bond depends u p o n only the types of the adjacent C-C bonds. A particular bond is more fully described by means of a notation in terms of the bond types of adjacent C-C bonds. The following diagram s h o w the types of the bonds in naphthalene. As an
2c(): 2
2
3 9 0 1 2
3 3
2
example of the notation the a-/3 bond is dcsignatedl2v3the 6-p bond (2,2), and the central b o d (3,3,3,3): Now it follows from the Vroelant and Daudel approximation that the bond order of a bond is related to the above notation. As there is a relation between bond order and length we are able t o construct Table I giving for (3) IIartnisnn, 2. Nalurhrschuny, A, IPeb., 1947. (4) Vroelaut and h i i d e l , Bull. 8or. ckim. France, 16, 36, 277 (1949).
0 . CHALVET AND R. DAUDEL
366
each bond (designated by the notation described above) the bond order and the bond l e ~ i g t h . ~ Penney's r-bond BondQ order
TABLE I Length
Bonda
Penney'a *-bond order
(2) 0.827 1.362 (2,3,3) 0.493 (3) 0.873 1.353 (2,3,4) 0.508 (1,l) 0.437 1.438 (2,4,4) 0.537 (1,2) 0.524 1.416 (3,3,3) 0.537 (1,3) 0.566 1.408 (3,3,4) 0.556 (2,2) 0.612 1.398 ( 3 , 4 , 4 ) 0.578 (2,3) 0.652 1.388 ( 2 , 2 , 3 , 3 ) 0.324 (2,4) 0.676 1.385 ( 2 , 2 , 3 , 4 ) 0.346 (3,3) 0.692 1.380 ( 2 , 2 , 4 , 4 ) 0.367 (3,4) 0.714 1.377 ( 2 , 3 , 3 , 3 ) 0.367 (4,4) 0.735 1.373 (2,3,3,4) 0.387 ( 1 , 2 , 3 ) 0.367 1.455 ( 2 , 3 , 4 , 4 ) 0.407 (1,2,4) 0,387 1.450 ( 2 , 4 , 4 , 4 ) 0.428 ( 1 , 3 , 3 ) 0.407 1.455 (3,3,3,3) 0.407 (1,3,4) 0.428 1.440 ( 3 , 3 , 3 , 4 ) 0.428 (1,4,4) 0.440 1.436 (3,3,4,4) 0.440 ( 2 , 2 , 2 ) 0.412 1.443 ( 3 , 4 , 4 , 4 ) 0.469 ' (2,2,3) 0.452 1.433 ( 4 , 4 , 4 , 4 ) 0.489 ( 2 , 2 , 4 ) 0.472 1.428 The numbers in parentheses give the bond C-C bonds adjacent to the bond in question.
Length
1.423 1.420 1.414 1.414 1.410 1.405 1.465 1.460 1.455 1.455 1.450 1.445 1.440 1.445 1.440 1.436 1.430 1.425
Vol. 56
orders are subject to small variations from one molecule to another. We can draw for each notation of bond, a curve giving the bond order (using the Coulson's definition or the Penney definition) as a function of its Pauling's double bond character. Figure I' shows the results for the notations (2,2), (2,3) and (3,3) and gives an idea of the accuracy of this method.
Calculation of the Bond Lengths in the Case of More Comdicated Molecules I n the case of more complicated molecules such as substituted hydrocarbons like toluene or heteroatomic conjugated molecule like quinoline, benzoic acid, Nz04and so on, no simple methods are presently known. Here we shall describe a rather complex method giving good determinations of the bond lengths in such molecules and taking into account a large number of factors. This method which is based on the molecular orbital theory, is an iterative one, and it is rather similar to the process used by Wheland and Mann,8 but is more systematical. In a first approximation the a integrals are taken types of as proportional t o the Pauling electronegativities of the neutral atoms,g and the fl integrals are assumed to be unity. TABLEI1 With such parameters the usual L.C.A.O. secular P equation is solved and the bond orders and the atomic charges are evaluated. From these values we must calculate the new a and the new p. The general method is as follow: Evaluation of Length0 the new a: In the first approximation the a inBond Notation Calcd. Obsd. tegrals have been taken proportional to the elec1.385 (3,3) 1.380 P tronegativities of the neutral atoms. 1.415 (2,3,4) 1.420 Q Since we know the approximative atomic charge, R (3,3,4,4) 1 .4;3G 1.43 and since there is a relation between the charge and S (4,4,4,4) 1.425 1.43 the electronegativity of an atom we are able t o evaluate a more accurate electronegativity and Using Table I, we can obtain immediately the length of a bond in a conjugated molecule when we consequently a more accurate a.Io The formulas used in such determinations are know its formula. Table I1 compares with experiments the results Eo = E f SQ (1) obtained in the coronene case. The agreement is where E is the electronegativity of a neutral atom good: That is to say, a conjugated hydrocarbon without substituent or heteroatom. and Eo the electronegativity for this atom with an If,is nevertheless possible to illcrease the accuracy effectiW charge &* 6 is a parameter which depends of the process. For a given notation, the bond On the atom'0 but is 0.3* 010
=
Eo/M
6 where M is a parameter usually near
(2)
Evalua3 tion of the new p : The bond orders obtained a t the -0.700; end of the first approximation are used t o estimate 8 a first value of the bond lengths as there is usually o,600$ a known relation between the bond orders and the interatomic distances.ll I n many cases (as in NzOJ it is necessary to correct the obtained lengths, 0.500$: taking into account the coulombic interaction due to the effective charges. l10;
-
1
1
I
I
0
0.25 0.33 0.5 0.66 1 Pauling's double bond charaoter8. Fig. 1.-The plots correspond to some molecules for which the bond orders have been yet rigorously calculated. ( 5 ) Vroelant and Daudel, C o m p t . rend., 288, 399 (1949). (6) Rohertson and White, J. Chem. Soc., 607 (1946); 358 (1947).
(7) C h l v e t . Compi. rend., 232, 165 (1951). ( 8 ) Whelapd and M a w , J . Chem. Plays., 17, 264 (1949). (9) As proposed by Mulliken, J . chim. phys., 46, 497, 675 (1949). (10) See about this question P. and R. Daudel, J . P h y s . , 7 , 12 (1946); Laforgue, J . chim. phys., 46, 568 (1949). (11) See Gordy, J . Chem. P h y s . , 16, 305 (1947); Bernstein, zbzd., 15, A68 (1947).
BONDLENGTHCALCULATIOK I N C'ONJUGATED MOLECULES
March, 1952
This may be done using the formula dQ,Q' = d 4- ( & , & ' / k d 2 ) (3) where dQpQlis the corrected distance, d the distance corresponding directly to the bond order obtained, Q and Q' the effective charges of the two atoms constituting the bond and k the force constant which is usually given as a function of the bond order.12 Since /3 is expressible as a function of the length13 we are able to evaluate the new p. From the new a, and the new /3, a new L.C.A.O. secular equation is solved and the total process repeated until the desired accuracy is obtained. Usually the first interation is sufficiently good to lead to convenient lengths. Such a process has been applied to the study of molecules like NzO4l4and NI- or X3CH3. l5 The results are now summarized and compared with the experimental determinations. Interatomic Distances in Nz04.14--Figure 2 gives the assumed distribution of the electrons in this molecule and Fig. 3 shows the bond orders and the charges obtained. Gordy's relations16have been used to determine the force constants and the interatomic distances from the bond orders.
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For the N-N bond we find 1.58 A 0.04 A. and thus explain the particularly large experimentally determined length (1.64 f 0.03 A.) Interatomic Distances in N3- and N3CH3.15Figures 4 and 5 show the results which are in good agreement with the experiments.
Electronic distrihut,ion assumed in Na-. -0.83 +O .6G -0.83 hi-------N-----&2,374 2 ,374 Charges and Bond Orders. Tbe corresponding length is 1.16 A. Fig. 4.
tn
c
# Fig. 2
0 '
\043
Fig. 3
Electronic distribution assumed in NaCH3. -0.37 4-0.52 -0.15 N _______ NN / 1,701 2,701 Hac Charges and bond orders. The corresponding lengths are respectively: A, 25 and A, 12 A. Fig. 5.
Thus we obtai? for the N-0 bond the distance: Finally as some recent works have shown it 1.205 f 0.0200 A. and the experimental value17 would be possible to improve such calculations is 1.17 f 0.03 A. introducing the /3 integrals for non-adjacent bondsla (12) Gordy, J . Chen. Phue., 14, 305 (1946). and the configuration intera~ti0n.l~ (13) See ref. 9. (14) Chalvet and Daudel, Compt. rend., 231, 855 (1950). (15) Bonrremay and Dsudel, ibid., 280, 2300 (1950). (16) See ref. 11 and 12. ( 1 7 ) Broadiey and Robertson, Nature, 164,915 (1949).
(18) See for example Fernandez, Cornpt. rend., 233, 56 (1951). (19) See for example Craig, Proc. Rou. SOC.(London), A200, 272, 401, 474,498 (1950); Coulaon and Fischer, Pkzl. MUD.,11,386 (1949); Sandorfy, Conapt. wd, 232, 2449 (1951).
