## 3 Results and Discussion
### TDDFT Results
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & \\ \hline \hline
**Expt** & **PBE0** & **M06-2X** & **\(\omega\)B97-XD** & **CAM-B3 LYP** & **LC-\(\omega\)HPBE** & **CAM-B3 LYP** & **Expt** & **M06-2X (eV)** \\ \hline
265 & 254 & 239 & 240 & 242 & 230 & 244 & 4.68 & 5.19 \\
388 & 455 & 352 & 342 & 349 & 279 & 349 & 3.20 & 3.52 \\
466 & 580 & 439 & 384 & 407 & 307 & 414 & 2.66 & 2.82 \\
482 & 582 & 436 & 379 & 406 & 308 & 412 & 2.57 & 2.84 \\
506 & 618 & 465 & 404 & 432 & 327 & 440 & 2.45 & 2.67 \\ \hline \hline \(a\) & 1.5236 & 0.9429 & 0.6593 & 0.7761 & 0.3837 & 0.8032 & & 1.136 \\ \(b\) & \(-\)144.23 & \(-\)11.146 & 71.977 & 40.161 & 128.53 & 33.349 & & \(-\)0.126 \\ R\({}^{2}\) & 0.9951 & 0.9948 & 0.9773 & 0.9927 & 0.9910 & 0.9946 & & 0.9982 \\ MAE & 80.8 & 35.2 & 71.6 & 54.2 & 131.2 & 49.6 & & 0.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental (see Table 1) and calculated \(\lambda_{\text{max}}\) data (nm) and statistical parameters of the linear fit \(y=a\cdot x+b\). MAE is the mean absolute error; R\({}^{2}\) the correlation coefficient. In the last two columns we report the data of the absorption maxima and statistical parameters in eV. values are obtained from systems differing because of the anion (Br\({}^{-}\), I\({}^{-}\) and PF\({}_{6}\)\({}^{-}\)), the cation (viologen and tetramethylviologen) and the solvent (dichloromethane and acetone). Therefore the protocol appears robust. From a practical point of view, in the visible range of wavelengths, once the absorption maximum has been calculated with the above TD-DFT computational protocol, it is sufficient to rescale the calculated value by adding about 35 nm in order to get the experimental result: \(\lambda_{\text{max}}\) = \(\lambda_{\text{max}}\)(TD-DFT) + 35.2 nm. The additive constant proposed, 35.2 nm, is obtained directly from the MAE reported in Table 3: since all calculated data are underestimated with respect to the experimental wavelength, but nevertheless well linearly correlated, adding the MAE brings the DFT predictions in closest match with experiments. In order to get more insights on the electronic transitions involved in the calculated absorption bands, we present an analysis of the orbitals involved. In Figure 4, we show the simulated lowest energy absorption band of four salts in dichloromethane together with the two electronic orbitals giving the largest contribution to the electronic transition(s) which are responsible for the absorption band itself. We limit our analysis to the four cases in CH\({}_{2}\)Cl\({}_{2}\) since the results for MVI\({}_{2}\) in acetone are qualitatively similar. In all cases, the largest contribution comes from a HOMO \(\rightarrow\) LUMO transition, the two orbitals also represented in Figure 4, these two being the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO). However, while for the iodide and bromide cases, the HOMO is represented by the p orbitals of the anion and the LUMO by a delocalized \(\pi\) system of the cation, for the hexafluorophosphate salt both HOMO and LUMO are localized on the cation, therefore in this case no CT band is predicted. This is, in fact, in agreement with the absence of color of the corresponding solutions and solid salts. In Figure 5, we show the orbital energies of the HOMO orbitals of the four representative samples in CH\({}_{2}\)Cl\({}_{2}\) (between \(-0.40\) and \(-0.25\) a.u. ), and the corresponding LUMO's (between \(-0.15\) and \(-0.05\) a.u.). It is clear from the figure that the tetramethyl substitution of the bipyridinium core destabilizes the LUMO level, while the HOMO level (being that of the iodide anions) is almost unaffected. This leads to a blue shift of the band. On the other hand, replacing the iodide anion with bromide, induces both a stabilization of the HOMO level (now localized on the bromide anion) as well as a destabilization of the LUMO level of the bipyridinium core leading to an even larger blue shift of the CT band.