Chapter 14, Problem 14.3P

Interpretation Introduction

Interpretation:

The complete MO picture and energy diagram for the allyl radical ${\text{(CH}}_{\text{2}}{\text{ = CH - CH}}_{\text{2}}^{\text{•}}\text{)}$ which has one unpaired electron is to be shown and the HOMO and LUMO are to be identified.

Concept introduction:

The overlap of the two atomic orbitals (AOs) results in the formation of two molecular orbitals (MOs). One of these, called the bonding MO, is formed as a result of constructive interaction. It is lower in energy than the original AOs. The other, called the antibonding MO, is formed as a result of destructive interaction. It is higher in energy than the original AOs. Antibonding MOs have a node (a nodal plane) situated between the two atoms. The overall character of the MO is determined by the number of bonding and antibonding interactions between the AOs. If the bonding interactions are more, the overall character is bonding. If the antibonding interactions are more, the overall character is antibonding. If they are equal or there are none, the MO is nonbonding.

Depending on whether the AOs overlap along the bond axis or away from the bond axis (sideways), the MOs are designated as $\text{σ}$ or $\text{π}$. The last, highest energy MO that is occupied is called HOMO, the highest occupied molecular orbital. The next MO, lowest energy empty orbital is called the LUMO, the lowest unoccupied molecular orbital. When the number of interacting AOs is odd, there also exists a nonbonding MO that has the same energy as the original AO.

In a molecule with double bonds, all bonding $\text{σ}$ or $\text{π}$ MOs are occupied, so that the highest energy $\text{π}$ bonding MO is the HOMO. The LUMO is either the nonbonding MO or the lowest energy antibonding $\text{π}$ MO. The $\text{σ}$ antibonding MOs are empty.

For a linear system of conjugated p orbitals, all nodal planes in a resulting $\text{π}$ MO must be positioned symmetrically about the center of that set of p orbitals.