**Hückel’s molecular orbital(HMO) theory.**

The π- electron system in conjugated planar molecules can be treated independently of the σ framework.

Aromaticity is usually described in molecular orbital (MO) terminology( For details on MO theory , refer post # 80-93) .The HMO theory is the theoretical basis for Hückel’s rule, which we studied in the previous post.

Erich Hückel developed this theory to explain the aromatic systems.This theory is the quantum mechanical approach to explain bonding in unsaturated hydrocarbons.

As seen in post 33, the Schrodinger’s equation can only be applied easily to simple systems. For complicated systems, one uses approximation methods namely- perturbation or variation method.**HMO theory helps to solve the Schrodinger’s equation for organic conjugated molecules.** It thus helps us to fully understand their structure.

In conjugated systems, it is the π electrons which predominantly dictate the properties of these compounds. It is so because , the π electrons that can be easily distorted , whereas the σ bonds are strongly bound. Thus, the σ electrons are more stable(less energy) as compared to the π electrons.These π electrons are therefore responsible for the chemical reactivity of these systems.

**σ-π separability** – HMO theory is based on the assumption that –

This is called the **σ-π separability.**

The two types of electrons move in different regions in space – σ electrons are in the plane of the molecule and the π electrons are located above and below this plane. Thus, we can treat the π framework separately because the σ skeleton of a planar conjugated system lies in the** nodal plane** of the π system and thus it does not interfere with the π system.

This means that the π and σ electrons can be treated separately !

This assumption greatly simplifies the process.*e.g.* Benzene has –

6σ bonds = 6 × 2 = 12 σ electrons and

3 π bonds = 3 × 2 = 6 π electrons.

However, for HMO we have to consider only the 6 π electrons. Thus, a 18 electron system is reduced merely to a 6 electron system!

- The convention is that
**π electrons occupy the p**._{z }orbitals - It is the
**π-system that is important**in determining the chemical and spectroscopic properties of conjugated polyenes(compounds having more than one double bond) and aromatic compounds. - This theory can be applied to organic molecules without the aid of computers.
- The HMO theory qualitatively describes π molecular orbitals in both cyclic and acyclic conjugated systems. Although , here we will be discussing aromatic compounds only.
### How to use HMO theory?

As studied in post 33, we have to use approximation methods to solve Schrodinger’s equation for complicated systems. Approximation methods give us an approximate answer – not exactly the true value , but a value close to the exact value. HMO theory uses the **variational approximation method** to **Linear Combination of Atomic Orbitals **(LCAO) – (refer posts 82 and 83)

**Variational approximation method .**

** **This is an approximation method to solve Schrodinger’s equation for ‘*not very simple*‘ systems. With this method one can find approximate wavefunctions such as molecular orbitals. In this method, we start with a **TRIAL WAVEFUNCTION** with similar energy (or similar parameters).This energy is always more than the actual energy of the wavefunction.

**E _{trial }_{Ψ} > E_{actual Ψ}**

Why do we assign a higher energy value to the trial wavefunction? Any system will always try to achieve the lowest energy state i.e it tries to minimize its energy. If we assume that the trial wavefunction has less energy than actual one, then it will mean that the system will try to adopt the trial state rather than the actual state. This assumption thus cannot be useful , as we are interested in the actual wavefunction and not the trial one(*the trial one is just imaginary wavefunction we have considered theoretically to do the calculations*).

The molecular orbital energies are expressed in terms of two parameters –

**α ⇒ Coulomb integral**– represents the energy of 2p atomic orbital before overlap. The MOs having less energy than α are bonding MOs and those having more energy than α are anti – bonding MOs.**β⇒ Resonance integral**– represents energy of stabilization due to π orbital overlap or it can be thought of π bonding energy between two adjacent atoms.This is due to stabilisation due to delocalisation of the electron between two orbitals.

The value of α is same for all carbon atoms in a molecule, irrespective of the neighbouring atoms.

**
**

**The usual sign convention is to let both α and β be negative numbers.**

In the next post we will learn some mathematics associated with this theory. Till then,

Be a perpetual student of life and keep learning…

Good day !

References and further reading –