Application to the Allyl System

The basis set size of three implies that we will need to solve a 3 x 3 secular equation. Hiickel conventions (b)-(e) tell us the value of each element in the secular equation, so that Eq. (4.21) is rendered as 

The use of the KrOnecker delta to define the overlap matrix ensures that E appears only in the diagonal elements of the determinant. Since this is a 3 x 3 determinant, it may be expanded using Cramer s rule as 

Since a and P are negative by definition, the lowest energy solution is a + /ip. To find the MO associated with this energy, we employ it in the set of linear Eqs. (4.20), together 

Some fairly trivial algebra reduces these equations to flz = J2ai 

While there are infinitely many values of ai, a, and aj. which satisfy Eq. (4.28), the requirement that the wave function be normalized provides a final constraint in the form of Eq. (4.11). The unique values satisfying both equations are 

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