# The BCL Method for DSSC: Basis and Applications +
## 5 The \(\mathbf{I_{T}}\) Parameter: A Measure of the Electronic Density on the Anchor Group
### Theoretical Background
#### 6.1.3 Amount of Charge Injected (F2)  
The Linker-Acceptor Interface  
The second index represents the amount of charge injected and is calculated employing an electrostatic model for modeling the linker-acceptor interface. After geometrical optimization of the triplet state of the complexes bounded to Ti(OH)\({}_{3}\), the atomic distances and Mulliken's charges displayed on Figure 24 were calculated. The values of both parameters remain the same for all the series, therefore appearing not to be influenced by the type of ligand. The carboxylate moiety contains two different interatomic distances between C and O, in accordance with a ketonic (1.25 A) and ether (1.31 A) types of bonding. As expected, the electron transfer process occurs between the acceptor and the donor. In this case, the donor is located on the moiety formed by the titanium atom with a positive charge, surrounded by four negative-charge oxygen atoms. The donor site is located in O\({}_{1}\) which is at 2.00 A from the acceptor. Its donor capacity is influenced by the net electrostatic potential created  
Figure 23: Scheme showing the composition of the F1 index. by the charge distribution composed by the two positive charges on Ti and C1 and the negative charge on O\({}_{2}\). Electron Injection  
The electronic transfer mechanism proposed contains three stages as shown in the upper part of Figure 25. The DLan-Ti(OH)\({}_{3}\) adduct is excited in the first step (a) resulting in a charge separation state, step (b), where an electron populates the SOMO level of the dye. Finally, in step (c), the electron migrates to the LUMO of the Ti(OH)\({}_{3}\) unit. Figure 24: Inter-atomic distances and Mulliken’s charge analysis on the [R-COO-Ti(OH)\({}_{3}\)] interface. Figure 25: Molecular orbitals involved in the electron injection: (**a**) Ground state; (**b**) thexi state; (**c**) charge separate state. H(D) = HOMO of the Donor, L(D) = LUMO of the Donor S(D) = SOMO of the Donor, \(\eta_{\rm D}\), \(\eta_{\rm A}\) = Hardness of Donor and Acceptor. At the bottom of Figure 25, the molecular orbitals involved in this process are displayed. In step (b), an energy gradient (\(\delta\epsilon\)) is created product of the difference between the chemical potential of the SOMO (\(\mu_{D}^{0}\)) and the LUMO (\(\mu_{A}^{0}\)) of the acceptor. This difference is minimized when an amount of electronic charge (\(\delta\)q) is transferred when both potentials equalize \(\mu_{A}=\mu_{D}\). \[\mu_{D} = \mu_{D}^{0}\ +2\eta_{D}\ \delta\text{q}+\text{V}_{\text{el1}} \tag{29}\] \[\mu_{A} = \mu_{A}^{0}\ +2\eta_{A}\ \delta\text{q}+\text{V}_{\text{el2}}\] (30) \[\delta\text{q} = \frac{\mu_{D}^{0}\ -\mu_{A}^{0}\ +V_{\text{el12}}}{2(\eta_{D} +\eta_{A})} \tag{31}\]  
In Equation (30), the \(\delta\)q value can be identified with the F2 index when suitable values are employed to evaluate it. In this sense, \(\mu_{D}^{0}\ =\varepsilon_{SOMO}\) and \(\mu_{A}^{0}\ =\varepsilon_{LUMO}\) can be employed. On the other hand, the electrostatic potential term is calculated as \(\text{V}_{\text{el1}}=\sum\limits_{1}\frac{\text{q}_{i}}{\text{r}_{iA}}\) from the charge distribution and the global hardness through the difference \(\varepsilon_{LUMO}-\varepsilon_{HOMO}\). \[\text{F2}=\frac{\varepsilon_{SOMO}(D)-\varepsilon_{LUMO}(\text{Ti}(\text{OH })_{3})+\text{V}_{\text{el12}}}{2\Big{(}\eta_{D}+\eta_{\text{Ti}(\text{OH})_{3 }}\Big{)}} \tag{32}\]  
It is interesting to note that for the case in which the performance of several types of donors in contact with a single acceptor is studied, two terms of Equation (31) remain constant, \(\text{V}_{\text{el12}}\) and \(\eta_{\text{Ti}(\text{OH})_{3}}\), whereby this equation can approximate and give origin to:  
\[\text{F2}\approx\frac{\delta\epsilon}{2\eta_{D}} \tag{33}\]  
Equation (33) provides a very interesting and unexpected result. It follows that the amount of charge injected depends on the energy difference between the SOMO of the excited dye and the LUMO of the acceptor group.