Question

Explain how symmetry arguments are used to construct molecular orbitals.

Answer 1

One of the main reasons for the intrusion of symmetry arguments into the culture of modern chemistry is undoubtedly the concept of the “Conservation of Orbital Symmetry,” as introduced by Woodward and Hoffmann some 30 years ago. In the language of quantum chemistry, orbitals are one-electron wave functions; the molecular orbitals of interest here are linear combinations of atomic orbitals, in the so-called LCAO approximation. Simple arguments based on the symmetry behavior of the molecular orbitals involved—i.e., the Woodward–Hoffmann rules, permit certain classes of chemical reactions to be classified as being symmetry-allowed or symmetry-forbidden.

As an example, we look at the molecular p orbitals (MOs) of butadiene, H₂C=CH—CH=CH₂, built from the atomic p orbitals (AOs) of the four carbon atoms; four AOs, four linear combinations, shown in Fig. ​Fig.6.6. The AO is antisymmetric, i.e., it changes its sign at the atomic nucleus, and the MOs are, correspondingly, antisymmetric with respect to the plane in which the four carbon atoms are assumed to lie. The symmetry behavior of the MOs is indicated by the different shading. It is seen that the MO Ψ₁, the one of lowest energy, is constructed so that the four AOs have the same sign; all the lobes on one side of the plane are positive, all those on the opposite side are negative. One condition on the AOs that has to be satisfied in constructing the MOs is that the latter must be mutually orthogonal. Each additional change of sign corrresponds to an increase in the orbital energy, so that the next MO Ψ₂ must have one change of sign, Ψ₃ two such changes, and Ψ₄ three. Each of the carbon atoms contributes one electron to the p system so there are four electrons to be accommodated. Since, according to the Pauli principle, only two electrons may be housed in the same orbital, the state of lowest energy is the one in which the two lowest energy orbitals Ψ₁ and Ψ₂ contain two electrons each.

Molecular orbitals of butadiene, built from the AOs of the four carbon atoms.

The chemical behavior of a molecule is determined largely by the properties of the highest lying occupied MO, the one in which the most loosely bound electrons are found (the so-called HOMO or “frontier orbital”). In our butadiene example this is Ψ₂. The symmetry properties of this orbital should therefore determine the course of the chemical reactions of butadiene, at least as long as the molecule is supposed to remain in its electronic ground state.

We now apply this kind of reasoning to follow the course of a simple reaction of butadiene, the ring-closure leading to cyclobutene. For identification purposes, we start with an unsymmetrically 1,4-disubstituted butadiene; call one of the substituents P, the other Q. If we now form a new bond between the outer carbon atoms to form a closed ring, there are two possible outcomes; the substituents can end up on the same side of the ring plane, or on opposite sides. On the basis of orbital symmetry arguments, which do we expect to occur? The lower drawing of Fig. shows the frontier orbital Ψ₂ in the molecular conformation in which the outer carbon atoms approach one another on the way to the ring-closure reaction. According to the Woodward–Hoffmann rules, based on orbital symmetry conservation, the orbital components at the two outer atoms must come together in such a way that lobes of the same sign match. This is only possible if the two end groups are rotated in the same sense—i.e., both clockwise or both anticlockwise. The term “conrotatory” is used to describe this kind of coupled motion. The two substituents P and Q must therefore end up on opposite sides of the ring plane. Thus, as long as the ring-closure reaction proceeds via the electronic ground state of the molecule, we expect to get the transisomer.

Conrotatory and disrotatory motions leading to different cyclization products of a disubstituted butadiene.

By absorption of a suitable photon, molecules can be made to exist in an electronically excited state involving the promotion of an electron from a low-lying occupied orbital to a higher-lying unoccupied one. In the butadiene example, light absorption could lead to promotion of an electron from Ψ₂ to Ψ₃. In this case Ψ₃ would become the frontier orbital and its symmetry behavior would be decisive in influencing the outcome of the ring-closure reaction. As seen from the upper drawing in Fig. ​Fig.7,7, the lobe-overlapping condition now requires that the two end-groups be rotated in opposite senses—i.e., one clockwise and the other anticlockwise (known as “disrotatory” motion). As a result, in a photochemically activated ring closure reaction, the end groups P and Q should end up on the same side of the ring plane. Thus, the thermally activated reaction is expected to lead to one isomer, the photochemically activated one to another.

This presentation is greatly simplified and it has ignored the fact that, for thermodynamic reasons, butadiene derivatives do not undergo ring-closure reactions to form cyclobutene derivatives. It would be an uphill reaction in energy terms. In the real world, it is the other way round; cyclobutene derivatives can be transformed into butadienes, both thermally and photochemically, and the two modes of reaction indeed lead to different products in accord with expectations based on the rules. Similar arguments can be applied to several types of chemical reactions: ring-closure and ring-opening reactions, skeletal rearrangements, and others. The rules that emerge from orbital symmetry arguments help to systematize and rationalize a vast amount of experimental results and are enormously useful in planning the synthesis of complex molecules and in overcoming the stereochemical problems that arise there. They have become an essential part of the intellectual equipment of the practicing chemist.