Perturbation theory (qualitative question)

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Homework Statement



How does the energy change (negative, positive or no change) in the HOMO-LUMO transition of a conjugated polyene where there are 5 double bonds when a nitrogen is substituted in the center of the chain? The substitution lowers the potential energy in the center of the box (everywhere else V(x)=0 for particle in a box).

When there are 6 double bonds, the opposite change happens. Why?

Homework Equations



E1=<ψ0|H1ψ0>

E(pertubed)= E0 + E1λ

The Attempt at a Solution



Ok, so if you look at the particle in a box ψ*ψ for n=5 and for n=6, the center of the n=5 is at the top of a peak, while for n=6 it is at a node (i.e. where the probability=0). I'm not sure how to use this info to say how the excitation energy would change.

I think it means that for n=6 there is no change because there is no probability of an electron being there so the substitution does not change the excitation energy. And for n=5 there is a decrease in potential energy so E1 is more negative and the gap would be larger? (or would a decrease in V(x) mean that the gap is smaller?)

Does any of that make sense? Again, I just need a qualitative answer, and it basically boils down to how E1 changes with the substitution.

EDIT: I noticed this thread seems to be related but I'm still not quite sure of the answer:
https://www.physicsforums.com/showthread.php?t=189681
 
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