Example Problem: Particle in a Box

Calculate the first p-p* transition of butadiene assuming that the molecule is a box of length 5.78 Å.

Solution: Butadiene is shown below.

There are four atomic p orbitals in the p-system of butadiene. There will be four molecular orbitals that will be formed from the linear combination of these atomic orbitals. The particle-in-the-box approach (also called the free electron model) assumes that each energy level corresponds to one of the p orbitals. We assume that two electrons will populate each MO. Thus, two electrons are placed in n=1 and two electrons in n=2 (HOMO). In this case the LUMO is n=3 and the p-p* transition is 2 à 3.

Using the particle in the box approach we have:

To convert this answer to eV use the conversion factor: 1.602 x 10^-19 J/eV

To convert eV to cm^-1 use the conversion factor: 8065.5 cm^-1/eV.