# The BCL Method for DSSC: Basis and Applications +
## 4 A Big Challenge: Understanding the Absorption Band of Ruthenium Dyes
### Results
#### 4.1.3 Transition Density Analisys (TDA)  
The forthcoming analysis is based on the existence of a transition state that originates in the Franck Condon excitations that populate a series of unoccupied molecular orbitals before thermal deactivation occurs. This set of LUMO orbitals are used to build the excited state electronic density (\(\varrho_{T}\)) and were generated when electrons from the ground state (\(\varrho_{\text{GS}}\)) are excited and jump to empty orbitals (\(\varrho_{l}\)) of the virtual state. These molecular orbitals combined together, form the transition density (\(\varrho_{T}\)) that contains the information of how electron density on the excited state is distributed on the molecular fragments (\(g_{k}\))  
\[\varrho_{T} = \sum_{\text{lum  
From de experimental absorption spectra the gamma parameter can be obtained, that relates to the energy absorbed by the complex in the visible region (400-600 nm)  
\[\Gamma=\int_{vi}^{vf}\varepsilon(v)dv \tag{7}\]  
From TDDFT calculations, the absorbed energy (\(E_{A}\)) can be calculated by the sum of single excitations (\(\Gamma_{\lambda}\)) of the product energy at which the excitation occurs, \(E_{\lambda}=h\
u\), multiplied by the oscillator strength (\(f_{\lambda}\))  
\[E_{A}=\sum_{\lambda}\Gamma_{\lambda}=\sum_{\lambda}f_{\lambda}E_{\lambda} \tag{8}\]  
On the other hand, an electronic transition occurs when an electron from the ground state is excited to a molecular orbital of the virtual state. Frequently it occurs that many pairs of HOMO-LUMO orbitals, and not only one, are excited simultaneously: the set of HOMO is identified by \(\rho_{GS}\) and the set of LUMO by \(\rho_{T}\).  
Figure 9a shows that at some specific wavelength the process \(\varrho_{GS}\ \rightarrow\ \varrho_{T}\) occurs and involves pairs of fragment groups from the ground state {\(g_{\alpha}\)} that are linked with a fragment of the excited state {\(g_{\beta}\)}. This gives rise to a set of correlated pairs of fragments represented by \(\omega(\alpha,\beta)\). Therefore, it is possible to define a set of basic processes containing specific combinations of molecular fragments, \(\omega_{0}(\alpha,\beta)=\omega_{0}^{\alpha\beta}\), as shown in Figure 9b.  
This figure shows that the electronic transitions MC and MB belong to the category of metal-to-ligand charge transfer, MLCT, and involve a charge transfer from the ruthenium atom (M) to the chromophoric ligand (MC) or to bipyridine (MB). The ligand to ligand (LL) transition corresponds to a charge transfer between the F-substituted fragment of the chromophoric ligand and bipyridine.  
Finally, Figure 9 shows a charge transfer process that occurs inside the chromophoric LNMR2 ligand, which is classified as an intraligand process (ILCT). If the intraligand  
Figure 9: (**a**) ground state density \(\rho_{GS}\), transition density \(\rho_{TS}\), \(\omega(\alpha,\beta)\) distribution function, {\(\alpha\) \(\beta\)} molecular orbitals. (**b**) Possible electronic transitions: MB: MLCT band to the bpy carboxylate ligand; MC: MLCT band to the chromophoric ligand; LL transition; ILCT: intraligand charge transfer.  
process does not show clear charge transfer character, it is named as IL, as would be the case of complexes with the LH ligand.  
From the absorption spectra an absorption occurring at wavelength \(\lambda\) can now be described as sum of single basic excitations according to,  
\[\Gamma_{\lambda}=\sum_{k}d_{k}\omega_{0}^{a\beta} \tag{9}\]