(5)
According to Kuhn's model, the "box" is the chain of conjugated double bonds with "walls" at the nitrogens. Thus, equation 4 can be used to predict the energies of the levels available to the pi electrons. The lowest energy (longest wavelength) transition corresponds to promotion of an electron from the highest filled level (n1) to the lowest vacant level (n2=n1+1). The value of n1 is determined by the Pauli Exclusion Principle, which limits the number of electrons in each level to two (with opposed spins). For a system with N pi electrons a total of N/2 levels is occupied, so that
(5)
For example, the value of N for dye #2 is 8 (one pi electron from each carbon, two from each nitrogen, minus one for the +1 charge).
Estimation of the effective box length is much more subject to interpretation. The problem is to decide on the locations of the "walls". Kuhn assumed that the chain ends extend one bond distance past each nitrogen. Thus, L may be taken as the sum of the bond lengths between the two nitrogens plus one additional bond distance at each end. Because of the delocalization the carbon-carbon bonds should all be equivalent with bond lengths of about 1.39 Angstroms, as in the benzene molecule. For the carbon-nitrogen distance the average of 1.47 Angstroms (C-N) and 1.28 Angstroms (C=N) may be used, and 2 x 1.47 Angstroms accounts for the extra distance at each end.
Although the positive charge cannot be delocalized onto the terminal benzene rings (Try writing resonance structures.), the presence of these groups causes the potential to rise less steeply than the step change specified in the original boundary conditions. Effectively this lengthens or shortens L by a constant amount a for a series of dyes of a given type (sets I and II in Figure 1). Thus,
(6)
where |a| must be less than one C-N bond length. If the DE's for a series of compounds are measured, the empirical parameter a may be determined by regression analysis. The sum
should be smallest for the best-fit a.
Equations 5 and 6 indicate that DE is proportional to N/L2. As the chain length increases both N and L increase. However, because L is squared, the net effect is a decrease in DE as the chain becomes longer.