114.Calculations of HMO theory(1).

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_iΦ_i + c_jΦ_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 1 becomes equation i as 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 !