In this post let us apply the LCAO method and approximations we studied earlier to a simple conjugated molecule and carry out the calculations for HMO theory.

Consider a simple conjugated molecule –

Each carbon atom is uniquely identified with a subscript ,which helps to demarcate the different atoms while solving equations. Here,we have labelled the two carbon atoms with subscripts i and j. The HMO theory only considers the π AOs. So,applying the LCAO method to the wavefunctions of the π orbitals on two two carbon atoms we get – (refer post 112)

Ψ = c_{i}Φ_{i} + c_{j}Φ_{j}

We know from post no 33 (Schrodinger equation) that ,

Substituting for ψ in the above equation –

We know that Φ_{i} and Φ_{j} are wavefunctions representing p_{z} atomic orbitals on the two carbon atoms. So, we know these quantities.However, we don’t know the energy E and the co efficients c_{i} and c_{j} . We further try to **get two equations** out of this one equation , to be able to form a matrix as follows –

Now , let us consider equation 1 first. We integrate that equation over the entire space as follows –

The terms c_{i} ,c_{j }and E are numbers and so they are out of the integral sign. After integration we get the following terms –

**∫Φ _{i }HΦ_{i }= H_{ii}.**

**∫Φ _{i }HΦ_{j }= H_{ij}.**

**∫Φ _{i }Φ_{j }= S_{ij}.**

**∫Φ _{i }Φ_{i }= S_{ii}.**

where H is the resonance integral and S is the overlap integral.

Thus,

equation 1becomesequation ias follows –

Similarly , we can convert equation 2 to equation ii –

So now we have **two simultaneous equations i and ii**. Let us see how we proceed further with these two equations in the next post. Till then ,

Be a perpetual student of life and keep learning…

Good day !