The Hückel p molecular orbitals of 1,3-butadiene
· Solving the secular determinant gives the molecular orbital energies.
· It also ensures that at these energies the secular equations will have a nontrivial solution (i.e. a solution other than all the coefficients in eq. 1 being 0).
We now expand the secular determinant:
Let y = x2, then:
This is a quadratic equation. The solution is:
a = 1, b=–3, c=1.
y = 2.62 or 0.38
x2 = 2.62 or 0.38
There are 4 solutions (remember ).
The 4 molecular orbital energies are: