Computational details
The calculations associated to the photoexcitation of the dyes were performed in the DFT and Time Depend-DFT (TD-DFT) framework [44] [45] implemented in GAUSSIAN 16 program [46]. Specifically, geometric optimizations, vibrational analysis and UV–Vis absorption spectra were carried out using the exchange-correlation functionals with Coulomb-attenuating method, CAM-B3LYP [47]. Also, the basis set 6-311 + G (d, p) [48] was used to describe carbon, hydrogen, and oxygen atoms. Thermal contributions (1 atm and 298 K) to the enthalpy and Gibbs energy were evaluated using the rigid rotor and harmonic oscillator model. The lowest 100 excitation energies have been calculated for the absorption spectra including acetone and ethanol as a solvent through the continuous solvation model (SMD) [49]. The results were described as a combination of contoured Gaussian functions with full width a half maximum height set to 0.3 eV centered at each excitation energy using the GaussSum program [50].
The TiO2 semiconductor surface was simulated using a 3x2x2 layer (TiO2)72 slab model from the TiO2 anatase crystal structure in its (101) facet, which has shown to be the most stable surface with high photocatalytic activity [51][52]. Thus, dye@(TiO2)72 systems were constructed with cell parameters of 11.22 x 42.78 x 36 Å. These parameters were chosen based on the size of the dyes and 10 Å empty space in c, which is added to avoid any interactions between the dummy replica when the slabs are periodically replicated.
After, calculations for the dye adsorption to TiO2 semiconductor were performed using the periodic density functional theory approach implemented in the Vienna Ab-initio Simulation Package, VASP 6.1 [53][54][55]. The exchange-correlation functional PBE[56] was chosen due to the reported cost-efficiency for many extended systems. The valence electrons were expanded on a set of plane wave bases with the Monkhorst-Pack grid[57] centered at the Γ-point (1 1 1) and a cutoff of 500 eV for the kinetic energy. The core electrons were described through the projector augmented wave (PAW) [58][59]. The models were fully optimized using ISIF = 3 which calculates using a stress gauge for the degrees of freedom in positions, cells, and volume, while the isolated dyes were used ISIF = 2, presents the same characteristics mentioned, but without degree of freedom in cells and volume. The geometry relaxation was considered convergent when the energy difference from the previous optimization step was less than 1 x 10− 4 eV, and the SCF tolerance was 1 x 10− 5 eV.
Theoretical analysis of electron injection step.
The adsorption energies were calculated through the total energy in the ground state of the optimized molecules.
$$\varDelta {E}_{ads}={E}_{dye-{{(TiO}_{2})}_{72}}- \left[{E}_{{{(TiO}_{2})}_{72}}+ {E}_{dye}\right]$$
1
Where \({E}_{dye-{{(TiO}_{2})}_{72}}\), \({E}_{{{(TiO}_{2})}_{72}}\) and \({E}_{dye}\) are the total energies calculated in their ground states.
The free energy change for the electron injection \((\varDelta {G}_{inj})\) was described by Matthews as a charge transfer reaction [4]. Based on this, Katoh [60] defined and relates with the injection efficiency (\(\eta )\), as:
$$\varDelta {G}_{injection}={E}_{OX}^{{dye}^{*}}-{E}_{CB}^{{TiO}_{2}}$$
2
Where \({E}_{OX}^{{dye}^{*}}\)is the oxidation potential of the excited dye and \({E}_{CB}^{{TiO}_{2}}\) is the reduction potential of the conduction band of the semiconductor. Thus, for the calculation of \({E}_{OX}^{{dye}^{*}}\)it is expressed through the oxidation potential of the ground state of the dye \({E}_{OX}^{dye}\) and the absorption energy associated with the maximum wavelength \({\lambda }_{max}\)of the photoinduced intramolecular charge transfer, assuming that the electron injection occurs from unrelaxed excited state.
$${E}_{OX}^{{dye}^{*}}={E}_{OX}^{dye}-{\lambda }_{max}$$
3
Moreover, \({E}_{OX}^{dye}\) is obtained as \(- {E}_{HOMO}\) and \({E}_{CB}^{{TiO}_{2}}\)as the \({E}_{CB }\)of TiO2. Finally, we also calculate the photon collection efficiency (LHE) as a photophysical parameter that consider the oscillator strength f of the wavelength \({\lambda }_{max}\)of the dye, so: