In the last post we got an overview of what matrices are and how they are used. Let us proceed further and see how this concept can be applied to Huckel theory.
We have already mentioned that this theory uses the LCAO method(refer post no 82 , post 83). In this method, when two or more atomic orbitals(AOs) constructively interfere, we get bonding molecular orbitals(BMOs) and when they interfere destructively, we get anti-bonding molecular orbitals(ABMOs).We add wavefunctions(ψ) for constructive interference and we substract them for destructive interference as shown in the above figure. We also know that , the BMO and ABMO wavefunctions can be represented as follows –
So, for n atoms we can construct BMOs by adding all AO wavefunctions(Φ) as follows –
We can extend this concept to the π atomic orbitals of a conjugated molecule using HMO theory. Consider a conjugated molecule as shown in the figure below-
It is conjugated as it has alternate double and single bonds. The above figure shows that there are double bonds between carbon atom labelled 1,2 and 3,4 and single bond between carbon atom 2,3.However, we know that there will be resonance in the molecule and so the double bond will be smeared over all carbon atoms as seen in the resonance hybrid below. pz orbitals on each carbon atom are responsible for creating the π electron cloud , above and below the plane of the molecule.
Applying the LCAO approximation to the π orbitals only(HMO theory only considers π AOs)we get –
The summation sign (Σ) indicates that we add all the quantities from i =1 to i =n i.e from 1st to nth carbon atom.
In the next post we will continue discussing this theory further. Till then,
Be a perpetual student of life and keep learning….
Good day !
References and further reading –