Hueckel Theory. We discuss the effect of replacing one carbon atom of cyclopentadien by oxygen to make the furan molecule. In this problem, either use software or a calculator to get eigenvalues and eigenvectors.

$($a$)$ Write down the Huckel Hamiltonian for cyclopentadien and for furan in terms of parameters ${}_C,{}_0$ and $,$ where we have assumed that ${}_{CC}={}_{CO}=.$

$($b$)$ Reasonable choices for these parameters are $=-75$kca$\frac{l}{m}ol,$ and ${}_0={}_C+\frac{}{2}.$ Justify why these choices are $($or are not$)$ reasonable.

$($c$)$ Solve for the eigenvalues of both systems. Make plots of the energy diagrams for cyclopentadien and furan side by side.

$($d$)$ Explain why we did or did not need to know the value of ${}_C$ to properly obtain the energy level diagrams above.