MICHAEL J. S. DEWAR AND NORAL. SABELLI
2310
Vol. 66
THE SPLIT p-ORBITAL (S.P.O.)METHOD. 111. RELATIONSHIP TO OTHEX 31.0.TREATMENTS AND APPLICATION TO BEXZENE, BUTADIENE, AND NAPHTHALENE’ BY MICHAEL J. S. DEWAR AKD XORA L. SABELLI George Herbert Jones Laboratory, University of Chicago, Chicago, Illinois hcaived June 8, 18618
Previous papers of this series have described a modification of the 1.c.a.o.m.0. method designed to allow for the effects of vertical correlation in unsaturated molecules. Here the relationship of the split p-orbita2 (s.p.0.) method to other m.0. treatments is discussed; it appears to differ from these only in the values ascribed to the individual two-electron integrals. h s.p.0. treatment of benzene is described, both with and without configurat.ioninteraction and with and without dflerential overlap. The results seem better than those given by other treatmenb, except for the semi-empirical method of Pariser and Parr. The amplest s.p.0. calculation, using sin le configurations with neglect of overlap, gives much better results than any other treatment using single configurations. T%e s.p.0. method also is applied successfully to butadiene and naphthalene.
I. Introduction shell configuration, where the 2n n-electrons occupy Previous papers of this series2 have described in pairs the n m.o.’s of lowest energy a modification of the usual m.0. procedure designed to allow for vertical correlation3 of electrons in Fii = JpiH’pi d r orbitals of p- or ?r-type symmetry. Before describing the application of this s.p.0. (split p-orbital) approximation to several polyatomic molecules we will first show its relationship to existing m.0. treatments. Consider a conjugated system of n atoms. In the simple Hiickel treatment the corresponding ?r-m.o.’s $m are written as linear combinations of the n 2p, a.o.’s pi of the individual atoms taking where HC is the core Hamiltonian and the (ij,kb) are the usual two-electron repulsion integrals. part In the m.0. treatment the wave function \k n for the set of 2n Ir-electrons is written (neglecting +in = C a m i p i (1) the normalizing factor) as the determinant i=l
+
The energies ( E ) of the m.o.’s are given by the secular equation det1Hij -
ESijI
=0
(2)
where Hij =: JpiHpj d r ; Sij = J p i p j d r (3) The Hamiltonian H i s not very clearly defined in this treatment so the H i j usually are treated as empirical parameters (see, however, Ruedenberg*). The coefficients u& in eq. 1 are given by the set of linear equations &i(Fii
- E,Sij)
=O
(i = 1, 2 . . . . .n) (4)
9 = det 1 #i$i$z&.
..#Bn I
(7) In the s.p.0. treatment, where it is assumed that the two electrons occupying a given ?r-m.0. are kept in separate lobes by their mutual repulsion, the corresponding (unnormalized) wave function 9’is given by \k‘ =
9lI(U(XiSi)) i
(8)
where zi, ~i are the z-coordinates of the electrons occupying the spin orbitals and U is a step function defined by
U ( x ) = 0 if
5
>0
f
= lifzthe value comesPonding to This shows that vertical correlation has a proa mean bond length of 1.40 A. (Table VI). nounced effect on the alternation of bond character in b ~ t a d i e n e . ~The bond order is still further reTABLE X duced if different p's are used for the central and EXCITATION ENERGIES ( E . v . ) OF STATES OF hj APHTHALENE (a) OBSERVED; ANrI CALCULATED BY (b) S.P.O., 8 = -1.77 terminal bonds. Table X shows that the s.p.0. method is equally E.v.; ( e ) POPLE; (d) PARISER; (e) F.E.M.o.;( f ) S.C.F.C.I. successful in the case of naphthalene, the agreement Exp. S.p.0. Poplea P-Pb F.e.m.o.c S.c.f.c.i.d State a b d f with experiment being very satisfactory. Whether lBm 3 , 99" 3 , 95 4, 02 ,;4 7 , 11 the method is slightly better than, or slightly worse lBzu 4,27e 4,86 4,65 4,49 4,54 8,37 than, the Pariser-Parr and Pople treatments de1B3U ( 5 . 6 3 e . / \ 6 , 1 0 6,13 5.94 5,83 pends on the assignment of the transitions a t 5.63 lBm \ 6 . 3 0 e . j5~ ,52 6.20 6.31 5,84 9,31 and 6.30 e.v. According to the s.p.0. results the 'Bzu 2.60' 3 . 2 7 3.09 2.18 assignment of these is opposite to that given by the 3 . 9 1 4 . 0 9 3 . 6 4 other methods, though the difference predicted by 'Bsu the Pople method is very small (0.07 e.v.). The 8B3" 3.95 4.40 4.02 'Bu
"4s
5.62 7.78 3 .64 4.79
5.93 7.32 3.60 4.00
6.20 7.87 3.92 4.62
5.91 8.29 3.96 4.61
L
.lo
'BlU 4.06 4 . 8 3 4.42 See ref. 40. See ref. 21. See ref. 18. See ref. 41. e See ref. 16, 42, 43. f This assignment remains undecided. 0 See ref. 44, 45. (32) R. S. Berry, J . Chem. Phys.. 86, 1660 (1957). (33) R. G. Parr and R. 8. Mulliken, ibid., 18, 1338 (1950). (34) A. Pullman, J . chim. phys., 61, 188 (1954). (35) J. W. Sidman, J . Chem. Phys., 27, 429 (1957). (36) W. C. Price and A. D. Welsh, Proc. Roy. Sac. (London), A174, 220 (1940).
(37) J. R.Platt and H. B. Klevens. Rev. Mod. Phys., 16, 182 (1944). (38) R. S. Mulliken, ibid., 14, 265 (1942). (39) D. F. Evans, J . Chem. Soc.. 1735 (1960). (40) J. A. Pople, Proc. Phys. 8oc. (London), A M , 81 (1985). (41) 9. Kolboe and A. Pullman, "Cslcul des Fonctions d'onde Moleculaire." Ed. CNRS, Paris, 1958. (42) H. B. Klevens and J. R. Platt, J . Chem. Phyd., 17,470 (1949). (43) H.Sponer and C. D. Cooper, ibid., 23, 646 (1955). Chem. Rev., 41, 401 (1947). (44) M. &ha, (45) D.F. Evans, J . Chem. Soc., 1351 (1957). (46) R. 9.Mulliken, Tetrahedron, 6, 68 (1959).
-1.STREITWIESER, JR.,AND I. SCHWAGER
2316
f.e.m.0. method predicts a still smaller difference in energy (0.01 e.v.> between these states. It is interesting that the s.p.0. treatment agrees with thc usual s.c.f. m.o. method (including configuration interaction) as regards the assignment of the transitions; the results given by thc latter method are, however, greatly in error. VI. Summary The main conclusions of this paper can be summarized as follows. (1) The s.p.0. treatment can be regarded as a modification of the usual s.c.f. method in which allowance is made for vertical correlation by adjustment of integrals. It could be regarded in this sense as a semi-empirical extension of the s.c.f. treatment along the lines pioneered by Pariser and Parr,' but with the integrals modified in a logical and self-consistent manner. The approach provides a further justification for the use of the s.p.0. method and further support for our contention2 that the neglect of non-orthogonality between s.p.0. functions and the core may not in practice have any serious consequences. (2) The results in this and the two preceding papers2 suggest that the s.p.0. method is a very promising one for calculating the properties of ronjugated systems. (3) The only one-configuration treatment that can compare with the s.p.0. method is the Poples approximation. In the case of hydrocarbons the two methods would be expected to give very similar results, for the following reason. Using a Coeppert-nIayerSklar potential and neglecting penetration integrals, eq. 19 becomes8
VOl. 66
Pople assumes, following Pariser and Parr,' that all the repulsion integrals have reduced values, corresponding in our system to "upper-lower" integrals. Since Qi = 1 for all atoms in an alternant hydrocarbon, if differential overlap is neglected,8 the Poplc and s.p.0. expressions for the diagonal elements in the F-matrix are identical for such compounds. The expressions for the off -diagonal elements differ, being given by
However, since bond orders do not vary much in aromatic compounds, and since the elements F,, are small for non-bonded atoms! the values for the two methods can be brought into near coincidence by using different values for p. The value appropriate to the s.p.0. treatment should of course be numerically smaller-as in fact it is. However this correspondence between the Pople and s.p.0. treatments applies only to alternant hydrocarbons; in the case of non-alternant hydrocarbons, or of compounds containing heteroatoms, the charge densities pi are no longer unity and the expressions for the diagonal elements of the Fmatrix differ. Preliminary results suggest that the s.p.0. method is significantly superior for heteroaromatic systems. Acknowledgment.--We are very grateful to Dr. L. C. Snyder of Bell Telephone Laboratories for helpful discussions and for his Molecular F , , = W i p t l/'*q,(ii,ii) (a, - l)(ii,jj) (25) Orbital Program, to Bell Telephone Laboratories for computational facilities, and to the National 3#i Science Foundation for support. N. L. S. thanks The corresponding s.p.0. equation (21) becomes the Consejo Nacional de Investigaciones Cientificas F , , = w?l, 'g,(ii,ii) (q, - l ) ( i i , j j ) (26) de Argentina for a Fellowship, for part of the period covered by this work. 3#a
+
+
+
A MOLECXLAR ORBITAL STLTDYOF THE POLSROGRAPHIC REDUCTIOS IS DIJIETHYLFORhfAMIDE OF UNSUBSTITUTED An'n METHYLSTTRSTITUTED AROMATIC HYDROCARBOXS' B Y A. STREITWIESER, JR.,* .4ND I. SCHWAGER Department of Clwriistry, Unii eruity of California, Berkeley, Cal. Received June 20, 1968
The half-wave potentials of a number of aromatic compounds have been determined and are found to fit the same type of correlation with the energies of the lowest vacant molwmlar orbitals in the HMO approximation established in other solvents. The deviation of biphenylene from this correlation can he accounted for in terms of "long bonds" but this esplanation fails when applied to 2,3-benxobiphenylene. Azulene and acepleiadylene do not obey the simple theory. The effect of methyl substituents is accounted for successfully in ternis of a combination of conjugative (hyperconjugation) and inductive influences.
Laitinen and Wawzonek found bhat phenylsubstituted olefins and acetylenes and aromatic polynuclear hydrocarbons are reduced a t hhe dropping mercury elect'rode in aqueous dioxane and
give reproducible half-wave potentials. The first reduction wave of these compounds corresponds to the approximately reversible addition of one or two electrons to the c o m p ~ u n d . ~ - 'Hence, ~ the half-
(1) This research was supported in p a r t by a n .4ir Force Grant, AFAFOSR-62-175. (2) Alfred P. Slosn Fellow, 1958-1962.
(3) H. A. Laitinen and S. Wawronek, J . Am. Chem. % e . , 64, 176.; (1942). (4) S. Wawzonek and H. A. Laitinen, ibid., 64, 2365 (1942).
