The energy level of the particle in a one-dimensional box are given by
$$E_n = \frac{n^2\hbar^2 \pi ^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}.\tag1$$
We see that the energy is quantised and dependent on the quantum number $n$. We also see that the energy difference between those levels gets larger with $n$. The only variable in this equation is the length of the box $L$. The larger the box the denser the states.
For an extended pi system like the ones in the dyes we make various simplification. First and foremost we reduce a multidimensional problem to one dimension. We also treat the barriers of the box as infinitely large potentials. Then we can only estimate the length of the pox. We are completely ignoring the fact that the molecule will have vibrations. And there are a couple of more assumption we make. All of them will influence the accuracy of the calculation. You should discuss these issues thoroughly and come to your own conclusions.
pH13 outlined a procedure that has been established in the literature, which neatly correlates the experimental findings with the calculations.[1] I personally find this method a little bit sketchy and not very intuitive at all. My first issue is the choice of the box itself; it seems rather arbitrary to me, as they decide to ignore the phenyl rings, add half a length of a bond, choose the average bond distance to be the one of benzene.
In this case, the box extends from nitrogen to nitrogen plus one-half of a bond length beyond each nitrogen. That is, $L = (p + 3)l$, where $p =$ the number of carbon atoms in the chain and $l$ is the average bond distance in the chain. The number of electrons in the chain, $N$, is $p + 3$.
[...]
[...] we let $l = 1.39 \times 10^{-8}~\mathrm{cm}$ (the bond distance in benzene) [...]
For a slightly simpler model I have estimated the length of the box to be $L=1.20~\mathrm{nm}=1.2\times10^{-9}$ and the number of π electrons to be $N= 22$.
With the reasonable assumption that orbitals are doubly occupied, we can hence find the the quantum number of the energy level of the HOMO $E_\mathrm{HOMO}$ to be $n_\mathrm{HOMO}=N/2$ and in analogy to that for $E_\mathrm{LUMO}$ as $n_\mathrm{LUMO}=N/2+1$. The energy of the photon to excite one electron from the HOMO to the LUMO is in first approximation it's difference $\Delta E_\mathrm{gap}$. Therefore we will find the formula you used after some transformations.
\begin{align}
\Delta E &= E_\mathrm{LUMO} - E_\mathrm{HOMO}\\
&= \frac{n_\mathrm{LUMO}^2 h^2}{8m_\mathrm{e}L^2}
- \frac{n_\mathrm{HOMO}^2 h^2}{8m_\mathrm{e}L^2}\\
&= \frac{h^2}{8m_\mathrm{e}L^2}\left[n_\mathrm{LUMO}^2
-n_\mathrm{HOMO}^2\right]\\
&= \frac{h^2}{8m_\mathrm{e}L^2}\left[\left(\frac{N}{2}\right)^2
-\left(\frac{N}{2}+1\right)^2 \right]\\
&= \frac{h^2}{8m_\mathrm{e}L^2}\left[N+1\right]\tag2\\
\Delta E &= \frac{hc}{\lambda}\\
\lambda &= \frac{8m_\mathrm{e}cL^2}{h(N+1)}\tag3
\end{align}
Plugging in the data, we find $\Delta E = 5.965~\mathrm{eV}$ and $\lambda=207~\mathrm{nm}$. Well, that's almost the same as you got, and that is very disappointing. Could it be, that there are a couple of approximations, that do not work well?
Farrell also admits that his approach is flawed:
Agreement between predicted transition energies and experimental energies, Table 1, is not good (although the agreement is rather surprising in view of the approximations made).
Therefore a parameter $\gamma$ which is adjusted to give a good fit to the experimental results is introduced. The length of his box becomes $L=p+3+\gamma$.
Elongating the conjugated bridge between the nitrogen atoms, lengthening the box, will result in a better fit, even without that adjustment. In other words, the less important the end groups (phenyl moieties) become, the better the simple model performs. That is not surprising at all.
The following table is reproduced from that paper (with slight modification). The number of carbons in the chain is $p$.
\begin{array}{lcccc}
p & \lambda_\mathrm{exp}/\mathrm{nm}
& \Delta E_{\mathrm{exp}}/\mathrm{eV}
& \Delta E_{\mathrm{th.}\, \gamma=0}/\mathrm{eV}
& \Delta E_{\mathrm{th.}\, \gamma=0.68}/\mathrm{eV}\\\hline
3 & 422 & 2.94 & 3.78 & 3.05\\
5 & 555 & 2.23 & 2.74 & 2.32\\
7 & 650 & 1.91 & 2.14 & 1.88\\
9 & 760 & 1.63 & 1.76 & 1.57\\
\end{array}
Conclusion
The failure of the model for short chains is not surprising. It will get better the longer the box is and the less influential the end moieties are. This system especially is well suited for discussions of such approximations and gain a basic understanding of quantum chemistry. Stay sceptical.
Reference
- John J. Farrel J Ed Chem 1985, 62, 351-352 [mirror]
Addendum
As conjugation increases the energy gap between HOMO and LUMO should decrease. With this relationship as N increases as does the energy gap. Makes no sense
It actually does. The more particles you fill in such a well, the higher the energy of the same ensemble gets. What makes the energy gap smaller and the states more dense is the extension of the actual box. The larger $L$ the closer are the states together. If the box size grows larger faster than the electrons you fill in it, then the gap shrinks. The gap is linear dependent on $N$, but inverse quadratic in $L$.