Huckel Molecular Orbital Theory

E. Huckel presented a simplified form of molecular orbital theory for 𝝅 system by introducing the following set of approximations-
1. σ, 𝝅 Separation
2. Free 𝝅 Electron Approximation
3. LCAO 𝝅 Molecular Orbital
4. The Energy of the HMO
5. The Secular Equation
6. Solving the Secular Equation.

1. σ, 𝝅 Separation

It is the first assumed that 𝝅-electrons in a conjugated molecule do not interact with σ-electrons. The complete molecular wave function being-
ψ = ψ_σ + ψ_𝝅
and the molecular energy-
E = E_σ + E_𝝅
This enables one to treat the 𝝅 molecular orbitals independently of the σ molecular orbitals.

2. Free 𝝅 Electron Approximation

The total 𝝅-electron wave function ψ_𝝅 is a single product of one electron functions (HMO) and not a determinant. The total energy of the 𝝅-electrons is then the sum of the one electron (HMO) energies.
For a system of n 𝝅-electrons in the ground state(with n even)
ψ_𝝅 = ψ₁(1) ψ₁(2) ψ₂(3) ψ₂(4) ... ψ_n/2(n − 1) ψ_n/2(n) -----Eq.1
where ψ_i is the ith HMO.
The energy is given by the relation-
E_𝝅 = n₁E₁ + n₂E₂ + ... + n_nE_n -----Eq.2
where, E_i is the energy of a single electron in ψ_i, the factors n_i may be 1 or 2 or 0 depending upon whether the HMO ψ_i is singly or double occupied or unoccupied.
Each ψ is an approximate solution of a one-electron Schrodinger equation,
Ĥ(t) ψ_i = E_i ψ_i -----Eq.3

3. LCAO 𝝅 Molecular Orbital

The equation-3 is never solved. Each HMO ψ_i is taken as a linear combination of the carbon 2p_z atomic orbitals-
ψ_i = a_i1P₁ + a_i2P₂ + ... + a_inP_n
or, ψ_i = ⁿ_r=1Σ a_ir P_r -----Eq.4
where, ψ_i is the ith HMO of the 𝝅 system, P_r is the 2p_z atomic orbital of the rth carbon atom and a_ir is the coefficient of the rth atomic orbital in the ith HMO. The HMO method is confined to carbon atoms only, orbitals of H atoms are not considered at all.

4. The Energy of the HMO

The energy of the HMO is calculated by the following formula-
E_i = ∫ ψ_iĤ ψ_i d𝜏 / ∫ ψ_i² d𝜏 -----Eq.4
assuming ψ_i to be real.

5. The Secular Equation

6. Solving the Secular Equation

Applications of Huckel Molecular Orbital(HMO) Theory

Source: Quantum Chemistry R.K.Prasad