513.001 Molecular and Solid State Physics

Hückel theory for the cyclopropene cation

The cation of cyclopropene is the smallest aromatic molecule. The Hückel model can be used to express three molecular orbitals in terms of the three p_z orbitals. Let α = <p_z1|H_MO|p_z1> be the matrix element with both <p_z1| and |p_z1> corresponding to the p_z orbital of the same atom. Let β = <p_z1|H_MO|p_z2> be the matrix element with <p_z1| and |p_z2> corresponding to the p_z orbitals of neighboring atoms.

How many electrons are in the system in total? How many electrons are in the molecular pi-system and have to be accounted for in Hueckel theory?

Set up the Hückel determinant and calculate the energies of the orbitals. What is the the degeneracy of the unoccupied orbtials?

The molecule can absorb a photon to promote one electron from the highest occupied molecular orbital to the lowest unoccupied molecular orbital. What is the wavelength of the photon (use β = 0.77 eV)?