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 = ciΦi + cjΦj
We know from post no 33 (Schrodinger equation) that ,
Substituting for ψ in the above equation –
We know that Φi and Φj are wavefunctions representing pz atomic orbitals on the two carbon atoms. So, we know these quantities.However, we don’t know the energy E and the co efficients ci and cj . 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 ci ,cj and E are numbers and so they are out of the integral sign. After integration we get the following terms –
∫Φi HΦi = Hii.
∫Φi HΦj = Hij.
∫Φi Φj = Sij.
∫Φi Φi = Sii.
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 !