5-5
51-
Figure 3. Energy diagram for interaction of starting wavefunctions and splitting of bonding and antibonding orbitals.
MO $111. As a,-3 is lower than a14 the amount of ~ 2 . 3in $1 will be greater than that of TI-4, whereas it will he smaller in $111. In the interference hetween a*,-, and a * l - 4 which gives the stabilized and destabilized MOs $11 and $rv, a similar apu nroach leads us to nredict that the contribution of a * ~will he greater in $11 and smaller in $1" than that of a*,-> From these considerations it follows that the central atomic coefficients will he greater than the terminal for $1 and $I", whereas they will be smaller for $11 and $111. Since the interaction energy between the starting wavefunctions is much greater than the splitting between a s - 3 and ~ ~ 2 .or3 a1.q and a * ~ .$111 ~ ,must be higher in energy than $11. These conclusions which are illustrated in Figure 3 are in agreement with Huckel or SCF calculations. Our approach to the relative contribution of the initial wavefunctions in the final MOs is of the same kind as that which is involved in many simple quantum explanation such as, for instance, the permutation of a and a levels in diatomic molecules ( 6 ) .It allows a clear understanding of the fact that the terminal coefficients are greater than the central in the frontier MOs and that an opposite situation exists for the two other MOs. Now if it is possible to explain the chemical properties of butadiene in a framework which involves explicitly the difference in magnitude of atomic coefficients, our knowledge of the origin of this difference added to this explanation will enable u s t o obtain a more complete understanding of the phenomenon by relating in a clear manner the chemical properties of this molecule to its topology. Let us first consider two examples to illustrate this fact. It is well known that the s-trans form of butadiene is more stable than the s-cis one. For both forms there is in $1 an attractive interaction between the AOs of 1 and 4 because thev are in phase, and in $11 there is a repulsive interaction because the AOs of 1 and 4 are out of phase. The magnitude of the repulsion energy involved in $11 is greater than the attractive one involved in $1 since the magnitude of the terminal atomic coefficients is smaller in $1 than In $11. It follows that there is a net repulsive interaction between atoms 1and 4. Of course this repulsion will he much stronger in the s-cis than in the s-trans isomer since the atoms 1and 4 are nearer in the s-cis butadiene. It is clear that the s-trans form is therefore more stable than the s-cis (Fig. 4). It is also possihle to explain in a similar way why the thermal ring closure of hutadiene to cyclohutene is conrotatory. In the conrotatory mode there is an in-phase overlap between the AOs of atoms 1 and 4 in $11 and an out-of-phase overlap for the same AOs in $I Since the magnitude of the terminal A 0
.
30 / Journal of Chemical Education
~
Figure 4. form.
Summary of interactions showing stronger stability of s-trans
Figure 5 .
Summary diagram or conrotatory and disrotatory modes
~
coefficients is greater in $11 than in $I, the attractive energy involved in $11 is more important than the repulsive energy in $1 (Fig. 5). In the disrotatory mode it is the opposite situation since the in-phase overlap in $1 is less important than the out-of-phase overlap occurring in $11. The conrotatory ring closure is therefore favored. The relative magnitude of atomic coefficients in the occupied MOs allows one to understand why the HOMO plays a privileged part in the two above examples. I t is also possible to find hy our qualitative approach the relative magnitude of the atomic coefficients in the MOs of s-cis and s-trans butadiene. As the distance between the atoms 1 and 4 is greater in the s-trans than in the s-cis butadiene, the splitting between the approximate wavefunctions TI-4 and will he smaller in the first case. On the other hand, since the distance between the atoms 2 and 3 is the same for both forms, the splitting between a % 3 and V'2-3 remains the same (Fig. 1).Thus the energy difference between the initial interacting wavefnnctions is smaller in the s-cis than in the s trans isomer because the amount of wavefunction mixing decreases with an increase in the energy gap. The mixing of the same symmetry starting wavefunctions will he smaller for s-trans than for s-cis butadiene and thus the difference in the magnitude between the terminal and central atomic coefficient will be greater for the s-trans. It is known that the molar extinction coefficients r associ$LU excitation which is greater for the ated with the $HO s-trans than for the s-cis butadiene is simply related to the transition dipole moment j~ which is given by
-
I;=
J ~ O ? + L U ~ V
kind:
.I
(i(i(~o) CULUI ?I + CXHO~CI~LLII ?Z + C:~(HO)C~(L.U) ?3 + C ~ ( H C O~I I L ?4 UI where iiis the position vector of the atom. In this sum the sign of the terminal contrihution is opposite to that of the central. Therefore if the central and terminal coefficients would he the same magnitude in $HO and $ ~ u t, would vanish. However, a s the terminal coefficients are greater than the central, the central contrihution will he smaller than the terminal. The greater this difference in magnitude, the stronger will be F (Fig. 6). Since the mixing in the starting wavefnnctions is stronger in the s-trans form, this leads to a greater difference in this case and thus t o a higher value o f t . Of course i acts in the same way a s the coefficients. We thank Pr. J. Soulier for discussions about the significance of the results of quantum computations. Literature Cited (11 I.uwry, T., Richardson. K.. "Mechanism and Theory in organic Chemislry.ii H a r p 1 Row. New York,1976. (21 Fleming I., "Frontier Orbitals and Organic Chemical Reactions," Wiley. NPWYmk, 1976.
c ~ ( ~C,(LU). ~ ) . Fi
i = 1 to4
Figure 6. Diagram showing the effect of the contribution of terminal and central po~itionstoward molar extinction coefficientvalues.
ii! Kinpman, C., "Cherniral Reaetasy and Reaeiiim Path; Wlley. New Ymk, 19'74, p. 55. ( 4 ) H k k e l , E . 2 Phyr., 70,204, (19311.76,628 11832). (51 Streitwieser. A,. "Molecular Orbital Theory for Organic Chemistry." Wiley, New Ymk. 1967, i ~39. . (6) Henri~nusseau,0..and Boulil,~..J. C H E M EDUC., 55,571 (19781.
Volume 58, Number 1, January 1981 / 31
