In the previous blog, we have learn how to set up the Hückel determinant for an aromatic molecule based on the topology of the pi-bonds. In this second part, we are going to learn how to calculate from the determinantal equation both the eigenvalues and the eigenvectors, corresponding to the orbital energy and orbital functions of the molecular system.

**EIGENVALUES FROM THE HÜCKEL DETERMINANT**

The calculation of the determinat gives a so-called **characteristic equation**. Namely a polynomial equation whose roots () are the eigenvalues of the system. As described in the previous article, the eigenvalues are related to the energy of the system by the relation Let work out the determinant for the allyl molecule. The determinant is given by

that is solved as

The polynomial has three root that can be easily found by rearranging it in giving and

We can now use these value to calculate the energy of the Hückel orbitals using the relation Therefore for

, we obtain , and

, we obtain , and

**REFERENCES**

J.P. Lowe Quantum Chemistry. 1993, Academic Press.