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:



The molecular orbital diagram for 1,3-butadiene