116.Aromaticity(9)- Calculations of HMO theory(3).

In the last post we computed Hückel calculations for two p orbitals in conjugation. Let us now try to figure out how to calculate energies for more than two p-orbitals.

The basics of these calculations remain the same and so one can easily find out energies for systems having more than two π orbitals in conjugation.

Consider a 3 π system such as an allyl radical–


As seen in the above figure, this system has three p-orbitals in conjugation on carbon atoms C1 , C2 and C3. Now if we have to construct a matrix, for this system ,  then it would look like –

1162According to Hückel‘s approximations –
For AOs on same carbon atoms i.e for C11 , C22 and C33 – the term used will be α-E.
For AOs on adjacent atoms i.e C12 , C21, C23  and C32 – the term used will be β.
For AOs on non-adjacent atoms i.e C13  and C31 – the matrix element will be zero.

Dividing by β and then substituting  (α-E / β) = x, we get – 


We know how to solve determinant for 3 × 3 matrix (post 111). The solution to above matrix will be –

1164The solution of the above matrix is –

x(x2-1)- 1(x-0) + 0 = 0.

x3 – 2x = 0.

On solving this equation, we get –

x = 0 , √2 , -√2.

The equation has three solutions as we are dealing with a 3×3 determinant.


(α-E / β) = 0, √2 , -√2.

(α-E / β) = 0


 α is the energy of the  p-orbital in no overlap situation. Thus, this represents a non bonding MO.

(α-E / β) = √2

  E = α – √2β

As β is
a negative number,
this energy corresponds to an antibonding MO.

 When x = -√2 

             E = α + √2β

This MO has the lowest energy and so is the bonding MO.



As seen in the above figure, the allyl radical is resonance stabilised, which means that it is more stable than its alkene counterpart. How does Huckel theory explain this stabilization?

We compare the π energy of the allyl radical (Eπ) with an ethylene molecule and an unconjugated p- orbital.


The delocalisation / resonance energy is NOT a measurable physical quantity. It is obtained by comparing a real molecule to a hypothetical one.

In the next post we will apply the HMO theory to benzene molecule. Till then,

Be a perpetual student of life and keep learning..

Good day !


References and further reading –



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