Homework Help: Perturbation theory (qualitative question)

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

E¹=<ψ⁰|H¹ψ⁰>

E(pertubed)= E⁰ + E¹λ

3. 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