# PIB wave functions and the effects of e- delocalization on UV absorption in conjugated systems

I'll try to keep this short and concise. What made me start thinking about this question was the material we have been covering regarding PIB wave functions in my quantum chemistry class, and material we have been covering regarding ozone absorption in my climate class. I would like to try to understand the relationship between the increased wave function domain that delocalization yields for conjugated pi electrons, and the effect that this has on the wavelengths of light that they can absorb. Like ozone, I understand that most other conjugated pi systems have strong UV absorption, so I would assume there is some correlation there.

Specifically though, I would like to know if this can be explained (at least to some extent) by a simple QM model like the 1D PIB, and if how electrons' wave functions are altered when their spacial domain is extended across the whole molecule by delocalization can explain UV absorption in conjugated pi systems.

You can use the PIB model as a very approximate way of thinking about energy levels in conjugated ($\pi$ electron) molecules. Absorption in triatomics is much more tricky, however, and usually a Walsh diagram is necessary.
The energy levels of a PIB vary as $E_n \approx (n/L)^2$ where n is the quantum number and L the length of the box, or conjugation. Take this to be sum of all conjugated bond lengths. Thus as L gets larger the energy levels become closer together. Secondly as the number of conjugated bonds increases the number of quantum levels filled also increases, as there are more electrons. The optical transition is from the highest filled level to the next one up (HOMO-LUMO), the energy difference is $E_{n+1}-E_n$ at each value of L. If you take the length of the box to be the number of bonds m of length a, then $L=ma$ it is easy to show that the absorption spectrum shifts to the red as the conjugation increases, although you should definitely not expect accurate values in terms of energies.