The HMO theory makes certain approximations.These approximations have proved to be very useful as they help simplify calculations and give accurate results. So, they are just accepted.
Hückel’s approximations
In this theory we are going to compute two matrices namely for –
i) The overlap integral and S_{ij} and
ii) The resonance integral H_{ij}.
i) The overlap intergral (S) – We have already studied what an overlap integral is in post # 83. Here , the overlap intergral , S_{ij,} refers to the extent of overlap between the AOs of two p_{z} orbitals .
If i and j are two carbon atoms under consideration, then –
- If i = j i.e we are considering the AO on same carbon atom then ,S_{ij} = 1 , as the overlap integral for an AO with itself is 1 for a normalised wavefunction. As studied in post # 83 ,normalization means integrating over the entire space of that orbital. The value 1 indicates that the overlap will be over the entire space.
- HMO theory makes an assumption ⇒ when, i≠j , then S_{ij} = 0, which means when we are considering two AOs on different carbon atoms , the overlap integral is zero. This is because there is no interaction between these two AOs, as they are far apart and the two AOs are said to be orthogonal(do not interfere with each other).
From the above information, we can conclude that S_{ij} is an identity matrix with all ones(1) on the diagonal and zeros(0) elsewhere. This approximation greatly reduces the complications in solving the math in HMO theory.
ii) The resonance integral, H_{ij} – This gives us the energy of the electron in the molecular orbital. We will consider the above two cases for this integral too.
- If i=j i.e we are considering only one AO (no overlap situation), H_{ii}= α . We have already seen in post # 110 that α is the energy of 2p orbital before overlap or in no overlap situation. This explains why H_{ii}= α.
- If i and j are adjacent carbon atoms (i= ±j e.g. -If we consider i= 2 then j has to be carbon atom #1 or 3i.e adjacent carbon) , then there will be overlap between them.
We already know that ,the resonance integral (β) , gives us the energy of π bonding .So, in this case , where i and j are adjacent atoms and there is overlap,
H_{ij}= β. - If i and j are apart(two or more bonds apart) then , the resonance integral β is negligible and it is taken as zero. So, H_{ij}= 0.
- We can summarize the above assumptions as follows –
^{ } |
i = j |
i≠j , i = j±1 ^{ } |
i≠j , i ≠ j±1 |
^{Overlap integral S}_{ij} |
1 |
0 | 0 |
^{Resonance integral H}_{ij} | α | β |
0 |
- In the next post we will discuss some more mathematical concepts required to understand the HMO theory and then we will proceed to apply it to specific molecules.
- Till then be a perpetual student of life and keep learning…
- Good day !