114.Aromaticity(7) -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 –

1142

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 ,

334

Substituting for ψ in the above equation –

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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 –

1144
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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 HΦ= Hii.

∫Φi HΦ= Hij.

∫Φi  Φ= Sij.

∫Φi  Φ= Sii.

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

Thus, equation 1 becomes equation i as follows –

1147.jpg

Similarly , we can convert equation 2 to equation ii – 

1148

 

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 !

 

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