Orbital interaction overlap integral

The orbital interactions are controlled by the overlap integrals (Scheme 9) and the energy gap between the orbitals (Scheme 10) ... 

The chemical reactions through cyclic transition states are controlled by the symmetry of the frontier orbitals [11]. At the symmetrical (Cs) six-membered ring transition state of Diels-Alder reaction between butadiene and ethylene, the HOMO of butadiene and the LUMO of ethylene (Scheme 18) are antisymmetric with respect to the reflection in the mirror plane (Scheme 24). The symmetry allows the frontier orbitals to have the same signs of the overlap integrals between the p-or-bital components at both reaction sites. The simultaneous interactions at the both sites promotes the frontier orbital interaction more than the interaction at one site of an acyclic transition state. This is also the case with interaction between the HOMO of ethylene and the LUMO of butadiene. The Diels-Alder reactions occur through the cyclic transition states in a concerted and stereospecific manner with retention of configuration of the reactants. 

The frontier orbital interaction is forbidden by the symmetry for the dimerization of ethylenes throngh the rectangular transition state. The HOMO is symmetric and the LUMO is antisymmetric (Scheme 25a). The overlap integrals have the opposite signs at the reaction sites. The overlap between the frontier orbitals is zero even if each overlap between the atomic p-orbitals increases. It follows that the dimerization cannot occur throngh the fonr-membered ring transition states in a concerted and stereospecfic manner. 

The first interaction has favorable orbital phase overlap for a concerted (2ns + vizs) reaction. The interaction integral y, Eqs. 3—6, for a concerted process would have a maximum value if the two molecules approached each other so that the reacting orbitals could overlap in the most efficient manner. The best geometry would involve a face-to-face reaction of the two reactant species. The stereochemical consequence of such a reaction would be specific retention of substituent relative geometries. 

Through-space interaction. Although the two TT-orbitals 7ra and 7Tb are not in conjugation, there exists a small but finite cross term B between them which, to a first approximation, will be proportional to their overlap integral Sab = (7ra 7Tb). 

Since the energies of the unperturbed orbitals are assumed to be independent of the rotational angle, only trends in overlap integrals need be considered in order to determine the relative stabilization of the staggered and eclipsed conformations. We now consider in detail the various MO interactions and their impact on conformational preference ... 

The appropriate interaction diagram is shown in Fig. 21. The energies of the unperturbed orbitals are independent of the degree of rotation about the C-C single bond. Consequently, we need only consider changes in the overlap integrals that accompany rotation. 

Fig. 54. Plot of AO overlap integrals in F20 vs. the FOF angle. AO overlap integrals 3, 6 and 7 are involved in the orbital interactions turned on by bending...

Notice that the overlap integral, S b, will depend on the position and orientation of the orbitals at the sites A and B, as does the intrinsic interaction integral, hAb- The minimum-energy solution is found by the variational method, which we use twice in Appendix A. Equation (3.6) is differentiated with respect to cA and with respect to cB, resulting in two linear equations which can be solved. Thus,... 

In summary, at least for the frontier orbitals where equation (3.45) is expected to be valid, a net destabilization depending in a complex manner on the square of the overlap integral and proportional to the sum of unperturbed orbital energies ensues from the interaction. 

Generalization 4 hAB k(eA + eB)SAB (k a Positive Constant, SAb Assumed Positive). The interaction matrix element is not precisely proportional to the overlap integral but the behavior with respect to distance and symmetry is essentially the same. In other words, hab will be zero by symmetry when Sab is zero by symmetry, but not otherwise. Also, hab decreases in magnitude as a function of increasing separation in much the same way as S. Thus, two orbitals will not interact if they behave differently toward local elements of symmetry. 

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