Intermolecular interactions in dye-sensitized solar cells: A ...
Intermolecular interactions in dye-sensitized solar cells: A computational modeling perspective
Mariachiara Pastore,* Filippo De Angelis * Computational Laboratory for Hybrid Organic Photovoltaics (CLHYO), Istituto CNR di Scienze e Tecnologie Molecolari, via Elce di Sotto 8, I-06123, Perugia, Italy. E-mail: [email protected], [email protected] RECEIVED DATE (to be automatically inserted after your manuscript is accepted if required according to the journal that you are submitting your paper to)
S.1 Models and computational details S1.1 Software Various quantum chemistry packages have been employed for the calculations presented in the manuscript: ADF1 and QUANTUM ESPRESSO2 were used for DFT geometry relaxation of dye@TiO2 complexes, while single point DFT and MP2 as well as TDDFT computations were carried out using Gaussian03 (G03)3 and Gaussian09 (G09).4 S1.2 TiO2 cluster models To model the TiO2 surface, we used two clusters, (TiO2)385, 6 and (TiO2)82,7 both obtained by appropriately “cutting” an anatase slab exposing the majority (101) surface.8 The considered clusters represent a reasonable tradeoff between accuracy and computational convenience and nicely reproduce the main electronic characteristics of TiO2 nanoparticles. Indeed, the lowest transition of the (TiO2)38 cluster calculated using time-dependent density functional theory (TDDFT) is 3.20 eV,9 in good agreement with the experimental bandgaps typical of TiO2 nanoparticles of a few nanometers in size (3.2-3.3 eV).10, 11 S1.3 Single-dye binding to TiO2 Organic dyes: geometry optimizations of the dye@(TiO2)38 systems were carried out in gas phase with the ADF program package1 employing the PBE exchange-correlation functional12 with a TZVP(DZVP) basis set for Ti (H, C, N, O, S). Ruthenium dyes: the N719@(TiO2)82 and Z907@(TiO2)82 structures waere optimized by the Car-Parrinello (CP) method13 using the PBE exchange-correlation functional,12 a plane-wave basis set and ultrasoft pseudopotentials.14, 15 S1.4 Multiple-dye adsorption on TiO2
Aggregation of indoline dyes: Using the (TiO2)82 slab we have modeled the dye-aggregation for both D102 and D149, initially selecting among all the possible dimeric arrangements the closest interacting ones with no explicit superposition of atomic structures.
After this preliminary
screening, a set of six configurations was selected for D102, while, due to the steric hindrance introduced by the presence of the second rhodanine moiety, only three of them were retained as possible candidates for the D149 molecule The nomenclature we use to label the dimer confirgurations is depicted in Figure S1: keeping fixed the position of the molecule placed in (0,0), each dimer is labeled by the (x,y) coordinates of the second molecule. Therefore, the six dimers of D102 examined in this study are labeled as (4,0), (-1,1), (-2,2), (0,2), (2,2) and (4,2); for D149 only the (0,2), (2,2) and (4,2) configurations are considered.
Figure S1: Graphical representation of the (101) Ti02 anatase surface. The scheme illustrates the convention adopted to indicate the relative positions of two molecules: each couple is labeled with the (x,y) coordinates of the second molecule, being the first one conventionally fixed in (0,0).
To take into account the effect of the geometry relaxation as a consequence of the interaction between the molecules upon their adsorption on the surface, for each configuration, we performed geometry optimizations of various dye dimers adsorbed onto TiO2 by means o the Car-Parrinello (CP) method13 using the PBE exchange-correlation functional,16 a plane-wave basis set and ultrasoft pseudopotentials.14, 15 To evaluate the relative stability of the various optimized configurations, we carried out single point MP2 as well as DFT calculations on the deprotonated dimers (i.e. after removing the TiO2 slab) in solution taking the optimized geometries of the dyes adsorbed onto TiO2.17 To test the performance of the dispersion corrected DFT, we also carried out for the protonated most stable dimer of (0,2)-D102 and (2,2)-D149, single point calculations in vacuo at MP2, B3LYP and B3LYP-D318, 19 level of theory with 6-31G* and 6-311++G** basis sets. The absorption spectrum of the D102 and D149 monomers and of each dimeric arrangement was computed by TDDFT with the hybrid B3LYP exchange-correlation functional.20 Both MP2 and TDDFT calculations were carried out with the 6-31G* basis set, taking into account solvation effects by means of the conductor-like polarizable continuum model (C-PCM)21, 22 as implemented in the Gaussian 03 package.3
Stark shifts: The simplest model to simulate the Stark shifts, possibly induced by the electron injection from the excited state of the dye into the semiconductor CB and by the presence of oxidized neighboring dyes, consists of two D149 molecules adsorbed on a TiO2 surface. This rather complex system is sufficiently realistic but yet tractable by high-level ab initio calculations. Exploiting the results of the work on the aggregation of D149 on TiO2, here we take the optimized molecular structure of the most stable dimeric arrangement on D149 adsorbed onto a (101) (TiO2)82 anatase slab, termed (2,2)-D149 (see computational details described above).
The absorption
spectra simulations have been carried out in acetonitrile solution on the optimized structure of the adsorbed deprotonated dimer obtained by removing the (TiO2)82
cluster. For the TDDFT
calculations, performed with the Gaussian03 code,3 the hybrid B3LYP exchange-correlation
functional20 and the standard 6-31G* basis set have been employed and the solvation effects included by means of the conductor-like polarizable continuum model (C-PCM).21, 22 To mimic the electrons injected into the oxide CB, that is into the titanium 3d orbitals, we put point charges, 23, 24 for a total charge of -1, -2 and -4, on the Ti atoms positions in the original (TiO2)82 cluster; therefore in each position we have a point charge of -1/82, -2/82 and -4/82 for TiO2(1), TiO2(2) and TiO2(3) respectively. For the C/TiO2(n) cases we computed the lowest 70 singlet excited states, while for the N/TiO2(n) ones the calculations were limited to the lowest 10 singlet excitations; the UV absorption spectra were then simulated using a Gaussian broadening with σ = 0.20 eV.
Aggregation of MP124 dye: For each dimeric configuration examined, the geometries of the dyes adsorbed onto the (TiO2)82 cluster were optimized by using the tight-binding approach implemented in the DFTB program.25 The relative stability of the optimized configurations was evaluated by single point MP2 calculations on the deprotonated dimers in solution, taking the optimized geometries of the dyes adsorbed onto TiO2; MP2 calculations were carried out with the 6-31G* basis set and the Gaussian03 code.3 Aggregation of Ru-dyes: The geometry optimizations of the Ru-dye dimer grafted onto the (TiO2)82 cluster was carried out in gas phase with the ADF program package1 employing the PBE exchange-correlation functional12 and a DZ basis set. The interaction energy of the dimer with respect to the non-interacting molecules was evaluated by performing single point B3LYP energy calculations, using G09,4 in vacuo with the DGDZVP basis set corrected by dispersion contributions (B3LYP-D3).18, 19, 26 S1.5 Co-sensitization of TiO2 Dye-different dye - FRET: the geometry of the donor/acceptor system directly determines the energy transfer rate, which is given by:
kF
1
R06
0 rA rD 6
(1)
where 0 is the lifetime of the donor excited state, R0 is the Föster radius, rD and rA are the position vectors of the donor and the acceptor respectively. The Föster radius, defined as the distance between the donor and the acceptor when the FRET has 50% probability, 26 can be obtained from the donor luminescence efficiency, QD, the overlap integral of the donor emission spectrum, FD, and the acceptor absorption spectrum A, and the orientation factor 2:
9000 ln(10) 2QD R 128 5 n 4 N A 6 0
F
D
( ) A ( ) 4 d (2)
with NA being the Avogadro’s number and n the dielectric constant of the medium. The dimensionless orientation factor 2 can vary from 0 to 4 and is given by:
2 (cos 3cos cos )2
(3)
where the angle , and (Figure S2) define the relative orientation between the two interacting dipole moments.27 For randomly oriented dipole moments, 2 is equal to 2/3 but for surfaceadsorbed dyes 2 can significantly deviate from the solution value.28
Figure S2. Definition of the angle , and determined by the relative orientation of the transition dipole moments of in donor and acceptor. Figure adapted from Ref. 27
To model the co-adsorption patterns, here we have extended and refined the approach previously set-up to investigate dye aggregation on titania.29 The strategy we employ can be summarized in the following steps: a. Determining the adsorption geometry and the energy level alignment of the AS02 and C106 dyes on TiO2 by performing full geometry optimizations of the dyes attached to TiO2 followed by single point calculations in solution. b. Keeping fixed the geometries of the TiO2-adsorbed dyes, selecting among all the possible 1:1, 1:2, 1:3 and 1:4 AS02:C106 co-adsorption arrangements the closest interacting ones with no explicit superposition of atomic structures on a grid of Ti atoms. c. Determining the most stable interacting configurations by performing single point B3LYP energy evaluation with the DGDZVP basis set corrected by dispersion contributions (B3LYP-D3)18, 19, 26 and simulating the optical response of the so-determined aggregates in solution by TDDFT.
d. For the 1:4 ERD:SD co-adsorption pattern, calculating for each AS02+C106 couple, composing the selected configuration, the donor-acceptor distance, the orientation parameter 2, given by the interaction between the transition dipole moments of AS02 and C106,27 the corrected Föster radius, R0 and finally, the corresponding energy transfer rate kFRET. The geometry optimizations of AS02@(TiO2)82 and C106@(TiO2)82 were carried out in gas phase with the ADF program package1 employing the Becke-Perdew exchange-correlation functional30, 31 with a TZP/DZ basis set for Ti, Zn, Ru/H, C, N, O, S. For the optimized dye@TiO 2 structures, the energy level alignments have been evaluated at B3LYP20/6-31G*-3-21G*(Ru) level of theory taking into account solvation effects by means of the conductor-like polarizable continuum model (C-PCM).21, 22 as implemented in the Gaussian 09 package.4 The TDDFT calculations on the selected configurations were carried out using the B3LYP functional and the DGDZVP basis. For AS02 we also calculated the S 1S0 transition carrying out the excited state geometry optimization in gas phase with the B3LYP functional and a 6-31G* basis set. Dye-coadsorbent: The ground state geometries of NKX@(TiO2)82 and CDCA@(TiO2)82 systems were optimized in gas phase with the ADF program package1 employing the Becke-Perdew exchange-correlation functional30, 31 with a TZP/DZ basis set for Ti/H, C, N, O, S. Keeping fixed the geometries of the TiO2-adsorbed molecules, we selected various among the co-adsorption arrangements the closest interacting ones with no explicit superposition of atomic structures on a grid of Ti atoms mapping the active Ti atoms on the anatase surface. The relative stability of the considered dimers was evaluated by performing single point B3LYP energy calculations in vacuo with the 6-31G* basis set corrected by dispersion contributions (B3LYP-D3).18, 19, 26
References: (1) te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T., Chemistry with ADF. J. Comp. Chem. 2001, 22, 931-967.
(2) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I., et al., QUANTUM ESPRESSO: A Modular and OpenSource Software Project for Quantum Simulations of Materials J. Phys.: Condens. Matter 2009, 21, 395502. (3) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A.; Vreven, J., T. ; Kudin, K. N.; Burant, J. C., et al. Gaussian 03, Revision B05; Gaussian Inc.: Pittsburgh PA, 2003. (4) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A., et al. Gaussian 09, Revision A.1; Gaussian, Inc.: Wallingford CT, 2009. (5) Persson, P.; Bergstrom, R.; Lunell, S., Quantum chemical study of photoinjection processes in dye-sensitized TiO2 nanoparticles. J. Phys. Chem. B 2000, 104, 10348-10351. (6) De Angelis, F.; Tilocca, A.; Selloni, A., Time-dependent DFT study of [Fe(CN)(6)](4-) sensitization of TiO2 nanoparticles. J. Am. Chem Soc. 2004, 126, 15024-15025. (7) De Angelis, F.; Fantacci, S.; Selloni, A.; Nazeeruddin, M. K.; Grätzel, M., First-Principles Modeling of the Adsorption Geometry and Electronic Structure of Ru(II) Dyes on Extended TiO2 Substrates for Dye-Sensitized Solar Cell Applications. J. Phys. Chem. C 2010, 114, 6054-6061. (8) Vittadini, A.; Selloni, A.; Rotzinger, F. P.; Grätzel, M., Structure and Energetics of Water Adsorbed at TiO2 Anatase \(101\) and \(001\) Surfaces. Phy. Rev. Lett. 1998, 81, 2954-2957. (9) De Angelis, F.; Tilocca, A.; Selloni, A., Time-Dependent DFT Study of [Fe(CN)6](4-) Sensitization of TiO2 Nanoparticles. J. Am. Chem Soc. 2004, 126, 15024-15025. (10) Weng, Y.-X.; Wang, Y.-Q.; Asbury, J. B.; Ghosh, H. N.; Lian, T., Back Electron Transfer from TiO2 Nanoparticles to FeIII(CN)63-: Origin of Non-Single-Exponential and Particle Size Independent Dynamics. J. Phys Chem. B 2000, 104, 93-104. (11) Khoudiakov, M.; Parise, A. R.; Brunschwig, B. S., Interfacial Electron Transfer in FeII(CN)64-−Sensitized TiO2 Nanoparticles: A Study of Direct Charge Injection by Electroabsorption Spectroscopy. J. Am. Chem Soc. 2003, 125, 4637-4642. (12) Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Physical Review Letters 1996, 77, 3865-3868 (13) Car, R.; Parrinello, M., Unified Approach for Molecular Dynamics and Density-Functional Theory. Phys. Rev. Lett. 1985, 55, 2471-2474 (14) Pasquarello, A.; Laasonen, K.; Car, R.; Lee, C.; Vanderbilt, D., Ab Initio Molecular Dynamics for d-Electron Systems: Liquid Copper at 1500 K. Phys. Rev. Lett. 1992, 69, 1982-1985. (15) Giannozzi, P.; Angelis, F. D.; Car, R., First-Principle Molecular Dynamics with Ultrasoft Pseudopotentials: Parallel Implementation and Application to Extended Bioinorganic Systems. J. Chem. Phys. 2004, 120, 5903-5915. (16) Perdew, J. P.; Burke, K.; Ernzerhof , M., Generalized Gradient Approximation Made Simple. Phy. Rev. Lett. 1996, 77, 3865-3868. (17) Zhao, Y.; Truhlar, D. G., Benchmark Databases for Nonbonded Interactions and Their Use To Test Density Functional Theory. J. Chem. Theor. Comp. 2005, 1, 415-432. (18) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H., A Consistent and Accurate ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu J. Chem. Phys. 2010, 132, 154104. (19) Grimme, S.; Ehrlich, S.; Goerigk, L., Effect of the Damping Function in Dispersion Corrected Density Functional Theory. J. Comp. Chem. 2011, 32, 1456-1465. (20) Becke, A. D., A New Mixing of Hartree--Fock and Local Density-Functional Theories. J. Chem. Phys. 1993, 98, 1372-1377. (21) Cossi, M.; Barone, V., Time-Dependent Density Functional Theory for Molecules in Liquid Solutions. J. Chem. Phys. 2001, 115, 4708-4717. (22) Cossi, M.; Rega, N.; Scalmani, G.; Barone, V., Structures, and Electronic Properties of Molecules in Solution with the C-PCM Solvation Model. J. Comp. Chem. 2003, 24, 669-681.
(23) Hall, G. G.; Smith, C. M., Fitting Electron-Densities of Molecules. Int. J. Quantum Chem. 1984, 88, 4926-4933. (24) Smith, C. M.; Hall, G. G., Approximation of Electron-Densities. Theor. Chem. Acc. 1986, 69, 63-69. (25) Aradi, B.; Hourahine, B.; Frauenheim, T., DFTB+, a Sparse Matrix-Based Implementation of the DFTB Method. J . Phys. Chem. A 2007, 111, 5678-5684. (26) Hoke, E. T.; Hardin, B. E.; McGehee, M. D., Modeling the Efficiency of Förster Resonant Energy Transfer from Energy Relay Dyes in Dye-Sensitized Solar Cells. Opt. Express 2010, 18, 3893-3904. (27) Mårtensson, J., Calculation of the Förster Orientation Factor for Donor-Acceptor Systems with One Chromophore of Threefold or Higher Symmetry: Zinc Porphyrin Chem. Phys. Lett. 1994, 229, 449-456. (28) Pastore, M.; De Angelis, F., First-Principles Computational Modeling of Fluorescence Resonance Energy Transfer in Co-Sensitized Dye Solar Cells. J. Phys. Chem. Lett. 2012, 3, 21462153. (29) Pastore, M.; De Angelis, F., Aggregation of Organic Dyes on TiO2 in Dye-Sensitized Solar Cells Models: An ab Initio Investigation. ACS nano 2010, 4, 556-562. (30) Becke, A. D., Density-functional exchange-energy approximation with correct asymptotic behaviour. Phys. Rev. A 1988, 38, 3098-3100. (31) Perdew, J. P., Density-Functional Approximation for the Correlation Energy of the Inhomogeneous Electron Gas. Phys. Rev. B 1986, 33, 8822-8824.
Mariachiara Pastore,* Filippo De Angelis * Computational Laboratory for Hybrid Organic Photovoltaics (CLHYO), Istituto CNR di Scienze e Tecnologie Molecolari, via Elce di Sotto 8, I-06123, Perugia, Italy. E-mail: [email protected], [email protected] RECEIVED DATE (to be automatically inserted after your manuscript is accepted if required according to the journal that you are submitting your paper to)
S.1 Models and computational details S1.1 Software Various quantum chemistry packages have been employed for the calculations presented in the manuscript: ADF1 and QUANTUM ESPRESSO2 were used for DFT geometry relaxation of dye@TiO2 complexes, while single point DFT and MP2 as well as TDDFT computations were carried out using Gaussian03 (G03)3 and Gaussian09 (G09).4 S1.2 TiO2 cluster models To model the TiO2 surface, we used two clusters, (TiO2)385, 6 and (TiO2)82,7 both obtained by appropriately “cutting” an anatase slab exposing the majority (101) surface.8 The considered clusters represent a reasonable tradeoff between accuracy and computational convenience and nicely reproduce the main electronic characteristics of TiO2 nanoparticles. Indeed, the lowest transition of the (TiO2)38 cluster calculated using time-dependent density functional theory (TDDFT) is 3.20 eV,9 in good agreement with the experimental bandgaps typical of TiO2 nanoparticles of a few nanometers in size (3.2-3.3 eV).10, 11 S1.3 Single-dye binding to TiO2 Organic dyes: geometry optimizations of the dye@(TiO2)38 systems were carried out in gas phase with the ADF program package1 employing the PBE exchange-correlation functional12 with a TZVP(DZVP) basis set for Ti (H, C, N, O, S). Ruthenium dyes: the N719@(TiO2)82 and Z907@(TiO2)82 structures waere optimized by the Car-Parrinello (CP) method13 using the PBE exchange-correlation functional,12 a plane-wave basis set and ultrasoft pseudopotentials.14, 15 S1.4 Multiple-dye adsorption on TiO2
Aggregation of indoline dyes: Using the (TiO2)82 slab we have modeled the dye-aggregation for both D102 and D149, initially selecting among all the possible dimeric arrangements the closest interacting ones with no explicit superposition of atomic structures.
After this preliminary
screening, a set of six configurations was selected for D102, while, due to the steric hindrance introduced by the presence of the second rhodanine moiety, only three of them were retained as possible candidates for the D149 molecule The nomenclature we use to label the dimer confirgurations is depicted in Figure S1: keeping fixed the position of the molecule placed in (0,0), each dimer is labeled by the (x,y) coordinates of the second molecule. Therefore, the six dimers of D102 examined in this study are labeled as (4,0), (-1,1), (-2,2), (0,2), (2,2) and (4,2); for D149 only the (0,2), (2,2) and (4,2) configurations are considered.
Figure S1: Graphical representation of the (101) Ti02 anatase surface. The scheme illustrates the convention adopted to indicate the relative positions of two molecules: each couple is labeled with the (x,y) coordinates of the second molecule, being the first one conventionally fixed in (0,0).
To take into account the effect of the geometry relaxation as a consequence of the interaction between the molecules upon their adsorption on the surface, for each configuration, we performed geometry optimizations of various dye dimers adsorbed onto TiO2 by means o the Car-Parrinello (CP) method13 using the PBE exchange-correlation functional,16 a plane-wave basis set and ultrasoft pseudopotentials.14, 15 To evaluate the relative stability of the various optimized configurations, we carried out single point MP2 as well as DFT calculations on the deprotonated dimers (i.e. after removing the TiO2 slab) in solution taking the optimized geometries of the dyes adsorbed onto TiO2.17 To test the performance of the dispersion corrected DFT, we also carried out for the protonated most stable dimer of (0,2)-D102 and (2,2)-D149, single point calculations in vacuo at MP2, B3LYP and B3LYP-D318, 19 level of theory with 6-31G* and 6-311++G** basis sets. The absorption spectrum of the D102 and D149 monomers and of each dimeric arrangement was computed by TDDFT with the hybrid B3LYP exchange-correlation functional.20 Both MP2 and TDDFT calculations were carried out with the 6-31G* basis set, taking into account solvation effects by means of the conductor-like polarizable continuum model (C-PCM)21, 22 as implemented in the Gaussian 03 package.3
Stark shifts: The simplest model to simulate the Stark shifts, possibly induced by the electron injection from the excited state of the dye into the semiconductor CB and by the presence of oxidized neighboring dyes, consists of two D149 molecules adsorbed on a TiO2 surface. This rather complex system is sufficiently realistic but yet tractable by high-level ab initio calculations. Exploiting the results of the work on the aggregation of D149 on TiO2, here we take the optimized molecular structure of the most stable dimeric arrangement on D149 adsorbed onto a (101) (TiO2)82 anatase slab, termed (2,2)-D149 (see computational details described above).
The absorption
spectra simulations have been carried out in acetonitrile solution on the optimized structure of the adsorbed deprotonated dimer obtained by removing the (TiO2)82
cluster. For the TDDFT
calculations, performed with the Gaussian03 code,3 the hybrid B3LYP exchange-correlation
functional20 and the standard 6-31G* basis set have been employed and the solvation effects included by means of the conductor-like polarizable continuum model (C-PCM).21, 22 To mimic the electrons injected into the oxide CB, that is into the titanium 3d orbitals, we put point charges, 23, 24 for a total charge of -1, -2 and -4, on the Ti atoms positions in the original (TiO2)82 cluster; therefore in each position we have a point charge of -1/82, -2/82 and -4/82 for TiO2(1), TiO2(2) and TiO2(3) respectively. For the C/TiO2(n) cases we computed the lowest 70 singlet excited states, while for the N/TiO2(n) ones the calculations were limited to the lowest 10 singlet excitations; the UV absorption spectra were then simulated using a Gaussian broadening with σ = 0.20 eV.
Aggregation of MP124 dye: For each dimeric configuration examined, the geometries of the dyes adsorbed onto the (TiO2)82 cluster were optimized by using the tight-binding approach implemented in the DFTB program.25 The relative stability of the optimized configurations was evaluated by single point MP2 calculations on the deprotonated dimers in solution, taking the optimized geometries of the dyes adsorbed onto TiO2; MP2 calculations were carried out with the 6-31G* basis set and the Gaussian03 code.3 Aggregation of Ru-dyes: The geometry optimizations of the Ru-dye dimer grafted onto the (TiO2)82 cluster was carried out in gas phase with the ADF program package1 employing the PBE exchange-correlation functional12 and a DZ basis set. The interaction energy of the dimer with respect to the non-interacting molecules was evaluated by performing single point B3LYP energy calculations, using G09,4 in vacuo with the DGDZVP basis set corrected by dispersion contributions (B3LYP-D3).18, 19, 26 S1.5 Co-sensitization of TiO2 Dye-different dye - FRET: the geometry of the donor/acceptor system directly determines the energy transfer rate, which is given by:
kF
1
R06
0 rA rD 6
(1)
where 0 is the lifetime of the donor excited state, R0 is the Föster radius, rD and rA are the position vectors of the donor and the acceptor respectively. The Föster radius, defined as the distance between the donor and the acceptor when the FRET has 50% probability, 26 can be obtained from the donor luminescence efficiency, QD, the overlap integral of the donor emission spectrum, FD, and the acceptor absorption spectrum A, and the orientation factor 2:
9000 ln(10) 2QD R 128 5 n 4 N A 6 0
F
D
( ) A ( ) 4 d (2)
with NA being the Avogadro’s number and n the dielectric constant of the medium. The dimensionless orientation factor 2 can vary from 0 to 4 and is given by:
2 (cos 3cos cos )2
(3)
where the angle , and (Figure S2) define the relative orientation between the two interacting dipole moments.27 For randomly oriented dipole moments, 2 is equal to 2/3 but for surfaceadsorbed dyes 2 can significantly deviate from the solution value.28
Figure S2. Definition of the angle , and determined by the relative orientation of the transition dipole moments of in donor and acceptor. Figure adapted from Ref. 27
To model the co-adsorption patterns, here we have extended and refined the approach previously set-up to investigate dye aggregation on titania.29 The strategy we employ can be summarized in the following steps: a. Determining the adsorption geometry and the energy level alignment of the AS02 and C106 dyes on TiO2 by performing full geometry optimizations of the dyes attached to TiO2 followed by single point calculations in solution. b. Keeping fixed the geometries of the TiO2-adsorbed dyes, selecting among all the possible 1:1, 1:2, 1:3 and 1:4 AS02:C106 co-adsorption arrangements the closest interacting ones with no explicit superposition of atomic structures on a grid of Ti atoms. c. Determining the most stable interacting configurations by performing single point B3LYP energy evaluation with the DGDZVP basis set corrected by dispersion contributions (B3LYP-D3)18, 19, 26 and simulating the optical response of the so-determined aggregates in solution by TDDFT.
d. For the 1:4 ERD:SD co-adsorption pattern, calculating for each AS02+C106 couple, composing the selected configuration, the donor-acceptor distance, the orientation parameter 2, given by the interaction between the transition dipole moments of AS02 and C106,27 the corrected Föster radius, R0 and finally, the corresponding energy transfer rate kFRET. The geometry optimizations of AS02@(TiO2)82 and C106@(TiO2)82 were carried out in gas phase with the ADF program package1 employing the Becke-Perdew exchange-correlation functional30, 31 with a TZP/DZ basis set for Ti, Zn, Ru/H, C, N, O, S. For the optimized dye@TiO 2 structures, the energy level alignments have been evaluated at B3LYP20/6-31G*-3-21G*(Ru) level of theory taking into account solvation effects by means of the conductor-like polarizable continuum model (C-PCM).21, 22 as implemented in the Gaussian 09 package.4 The TDDFT calculations on the selected configurations were carried out using the B3LYP functional and the DGDZVP basis. For AS02 we also calculated the S 1S0 transition carrying out the excited state geometry optimization in gas phase with the B3LYP functional and a 6-31G* basis set. Dye-coadsorbent: The ground state geometries of NKX@(TiO2)82 and CDCA@(TiO2)82 systems were optimized in gas phase with the ADF program package1 employing the Becke-Perdew exchange-correlation functional30, 31 with a TZP/DZ basis set for Ti/H, C, N, O, S. Keeping fixed the geometries of the TiO2-adsorbed molecules, we selected various among the co-adsorption arrangements the closest interacting ones with no explicit superposition of atomic structures on a grid of Ti atoms mapping the active Ti atoms on the anatase surface. The relative stability of the considered dimers was evaluated by performing single point B3LYP energy calculations in vacuo with the 6-31G* basis set corrected by dispersion contributions (B3LYP-D3).18, 19, 26
References: (1) te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T., Chemistry with ADF. J. Comp. Chem. 2001, 22, 931-967.
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