# 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} = ψ_{1}(1) ψ_{1}(2) ψ_{2}(3) ψ_{2}(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} = n_{1}E_{1} + n_{2}E_{2} + ... + n_{n}E_{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_{i1}P_{1} + a_{i2}P_{2} + ... + a_{in}P_{n}

or, ψ_{i} = ^{n}_{r=1}Σ 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_{i}Ĥ 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