**P24.23** The allyl cation CH₂=CH—CH₂⁺ has a delocalized π network that can be described by the Hückel method. Derive the MO energy levels of this species and place the electrons in the levels appropriate for the ground state. Using the butadiene MOs as an example, sketch what you would expect the MOs to look like. Classify the MOs as bonding, antibonding, or nonbonding.

Huckel's method:- Huckel's method is used to calculate the molecular orbitals as a linear combination of atomic orbitals. It is used to determine the shape of π orbitals in a conjugated system. Calculation of MO of allyl cation by secular determinant:- The secular determinant is the method used to calculate the MO of the compound. The no of pi electrons = to the no of carbon atoms. So, the order of a determinant = n×n where, n= no of carbon atoms. in case of allyl cation, the structure is CH2=CH-CH2+ So, n =3. order will be 3×3 =9. So, the secular determinant will be Where, Hii=αii =columb integral overlap integral = β=Sij ( i≠j) (β= when bond is present between Ci and Cj. Sij =0 when i=j. Sij =1 when i≠j. Considering all of this, the secular determinant can be written as, H11-ES11H12-ES12H12-ES12H21-ES21H22-ES22H23-ES23H31-ES31H32-ES32H33-ES33=0 Putting all the proper Columb and overlap integral, we can write, α-Eβ0βα-Eβ0βα-E=0 Let, α-Eβ=x dividing everyone by β inside the determinant, α-Eβ101α-Eβ101α-Eβ=0x101x101x=0x(x2-1)-1x=0x3-x-x =0x3-2x=0xx2-2 =0so, x=0 ,±2 so,α-Eβ=0, or, E2=0 ....1)α-Eβ=+2E1= α+β2 .....2)α-Eβ=-2E3= α-β2 .....3) These are the 3 energy values. The value of β=-ve. So, equation 1 will e the lowest energy state and 3 will be the highest energy state. MO diagram:-