Why the Particle-in-a-Box Model Works Well for Cyanine Dyes but Not for Conjugated Polyenes

We investigate why the particle-in-a-box (PB) model works well for calculating the absorption wavelengths of cyanine dyes and why it does not work for conjugated polyenes. The PB model is immensely useful in the classroom, but owing to its highly approximate character there is little reason to expect that it can yield quantitative agreement with experimental data. However, we show that a more realistic behavior of the potential energy leaves the highest occupied and the lowest unoccupied PB energy levels unmodified as long as the variations in the potential are not too large (i.e., as long as first-order perturbation theory is valid). The result explains the success of the PB model in modeling absorption wavelengths of conjugated delocalized chains with small bond-length alternation such as cyanine dyes. In a next step, a suitable perturbation is applied to model conjugated polyenes that exhibit pronounced bond-length alternation and have a different behavior of the longest wavelength absorption as a function of chain length. In this case, the perturbation directly affects the highest occupied PB energy levels in a way that agrees with experimental observations.