First-Principle Determination of Electronic Coupling and Prediction of

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First-principle Determination of Electronic Coupling and Prediction of Charge Recombination Rates in Dye-sensitized Solar Cells Hsien-Hsin Chou, Chou-Hsun Yang, Jiann T'suen Lin, and Chao-Ping Hsu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b08553 • Publication Date (Web): 14 Dec 2016 Downloaded from http://pubs.acs.org on December 23, 2016

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The Journal of Physical Chemistry

First-principle

Determination of

Electronic

Coupling

and

Prediction of Charge Recombination Rates in Dye-sensitized Solar Cells§ Hsien-Hsin Chou,† Chou-Hsun Yang,‡ Jiann T’suen Lin* and Chao-Ping Hsu* Institute of Chemistry, Academia Sinica, Taipei 115, Taiwan (R.O.C.)

E-mail: [email protected] (Prof. Chao-Ping Hsu), [email protected] (Prof. Jiann T’suen Lin)

ABSTRACT: In the study of dye-sensitized solar cells (DSCs), developing computational predictions of key properties of the dyes is highly desirable for identifying promising candidates. In this work, we report first-principle–based, theoretically estimated chargerecombination (CR) rates from TiO2 to the cationic dye for 2 pairs of organic dye molecules. A recently developed multi-state fragment charge difference (msFCD) scheme, together with long-range–corrected time-dependent density functional theory, was used to calculate the electronic coupling. The msFCD scheme removes the local excitation components in the charge-transfer states, and generates acceptable diabatic states. The range-separated ωPBE and BNL functionals were useful for the charge-transfer problem, and results were largely independent of TiO2–dye binding modes. The rates obtained for CR to oxidized dye (CRD) followed a trend similar to experimental results. In general, a difference in the reorganization 1 `

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energy (λ) and the free energy (∆G0) had a large effect on electron transfer rates. However, electronic coupling strength could also have a dominant role over (λ+∆G0) in the CRD rate. We report a generally applicable ab initio approach to predict CRD rate and explored the potential roles of coupling factors in the performance of DSCs.

1. INTRODUCTION In dye-sensitized solar cells (DSCs), multiple electron transfer (ET) processes take place between the dye, the electrolyte and the TiO2 semiconductor. When a dye absorbs sunlight and becomes photoexcited, an electron is injected in the conduction band of TiO2. The injection rate is often very fast (typically picoseconds or faster),1 with high quantum efficiency. The cationic dye is regenerated by the electrolyte, which is further reduced by the cathode. The overall photon-to-current conversion efficiency will greatly decrease if the ET rates interconnecting these processes are inefficient or unbalanced.1-6 In additional to various deactivation processes of the dye, the loss of DSCs also arises from charge recombination (CR) of the TiO2 electron to the electrolyte (CRE) or to the oxidized dye (CRD).1, 3-4, 7-8 With the great potential of harvesting light, both outdoor solar or indoor ambient light, DSCs have drawn a very broad and intense interest in experimental studies. To date, solar cells with highest conversion efficiencies of about 13%9-10 and up to about 20%11 have been obtained under 1 Sun standard condition (100 mW cm-2) and 350 lux LED dim light (104.3 µW cm-2), respectively. Theoretical computations offering fundamental understanding, such as rates of elementary processes, are of great importance in understanding the factors involved in DSC performance.12-13 The computational approach can also delineate various complicated factors in linking the molecular structures to the electronic performance.

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In finding factors contributing to the loss of efficiencies, much attention has been paid to the suppression of CR processes.14-23 According to the Marcus theory,24-25 the rate of an ET is expressed as =

ℏ

| |

exp

,

(1)

where HRP is the electronic coupling, λ is the reorganization energy, and ∆G0 is the Gibbs free energy difference for the reaction. The exponential function in Eq. (1) is essentially a Franck-Condon factor accounting for the effects of nuclear motion.26-28 For an exothermic reaction, when (λ+∆G0) becomes negative, it reaches the inverted region, and the ET rate decreases as ∆G0 becomes more negative. The value of (λ+∆G0) generally dominates the reaction rate because it is in the exponential function. As seen in several reports of dipolar organic dyes, in a set of structurally similar dye molecules with similar reorganization energies, oxidative potentials, and even molecular dipoles, the measured CRD rates are often similar.14, 16-18, 22 However, given structurally similar molecules, the estimation of electron transfer rates judged solely by (λ+∆G0) could be misleading. One potential factor is the electronic coupling (HRP) in Eq. (1). The couplings are likely affected by the length, structure and modification of the π-linking moiety in the dye molecules. The π-linker is also a popular site for molecular design for desirable CRD couplings. Therefore, reliable estimation of coupling strength by a theoretical approach is important. With quantum chemical computation, the electronic coupling (HRP) for the CRD process can be calculated in the diabatic representation: = Ψ | |Ψ ",

(2)

where H is the electronic Hamiltonian of the system and Ψ and Ψ are the wave functions of the 2 diabatic states (Figure 1). 3 `



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Theoretical modeling of the CRD process for DSC is not seen frequently.19-21 CRD was previously studied with a tight-binding model for the conduction band of TiO2, and the coupling was estimated with the HOMO of the dye, essentially a one-electron model.21 Firstprinciple dynamic simulation was used in one study: both charge injection and CR for DSCs were reported, but the electronic coupling for CR was treated with a simple empirical distance-dependence expression.23 A first-principle electronic coupling avoids the potential problems in these assumptions and allows for better predicting power for molecular designs. In the present work, we further tested the use of long-range corrected29-32 (LC) timedependent density functional theory (TDDFT) because LC functionals are known to produce good charge-transfer (CT) excitation and electronic couplings.31-35 We used the multi-state variant of the fragment charge difference (FCD) scheme (msFCD)36 to calculate the coupling because it can remove erroneous local excited components in the diabatic states.37-38 With good-quality estimation for HRP, we studied the effect of coupling on the CRD process. 2. THEORY AND METHODS 2.1. Building models. As shown in Chart 1, the dye molecules included in this study are MZGB1, MZGB2, FTT and MCT, representing previously studied dyes 1, 2, FTT-1 and MCT-1, respectively,15, 39 with all aliphatic chains replaced by methyl groups. These molecules belong to D-π-A'-A type dyes, where "D" represents aryl amines, π is the conjugated aromatics, "A" is a cyanoacrylic acid anchor, and A' is the benzothiadiazole (as in MZGB dyes)15 or thienothiophene (as in TT dyes)39 group (Chart 1). The organic dye molecules in their neutral or oxidized form were optimized at the B3LYP/6-31G* level in a gas phase without restriction. The model we developed consists of an anatase titania cluster and a single dipolar organic dye, in configurations as shown in Charts 2 and 3. The geometry of the anatase titania cluster (TiO2)9 was as previously 4 `


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reported,40-41 which is a sandwich structure containing 3 (TiO2)3 layers. Each (TiO2)3 layer is bridged by 4 oxygen atoms, forming the anatase architecture. (TiO2)9 is shown to be large enough to reproduce the absorption spectra of the TiO2-dye system, with TDDFT calculations.41 The combined (TiO2)9-dye geometries were obtained by joining these 2 parts together, followed by optimization at the B3LYP/6-31G* level, with CRENBL pseudopotential for Ti atoms. For each of the (TiO2)9−dye complexes, we simulated 4 different chemically bonded models: the dye molecule anchors to the titania cluster by chelating with one Ti atom or bridging 2 Ti atoms on the surface layer. After anchoring to the titania cluster, the proton from cyanoacrylic acid is attached to oxygen on the surface layer near the carboxylate ligand (the chelate-Hsurf and bridge-Hsurf models, Chart 3) or on a further site (the chelate and bridge models). A summary of important bond lengths for the TiO2–dye system in this study is given for TT dyes (Table S2 in Supporting Information). We also studied physically absorbed models in which the proton stays with the organic dye molecules and the OH group is attached to an O atom in TiO2, but the electronic couplings of these cases are very low. Therefore, we considered chemisorb models in the present work. 2.2. The choice of density functionals. The use of DFT for properties such as CT excitation energies33 can be problematic because of the self-interaction errors in frequently used exchange-correlation potentials. LC functionals have great potential here since improved prediction of CT state energies31-32, 34 and CT couplings35 were reported. We used the popular B3LYP functional and 2 LC functionals, ωPBE29-32 and BNL,42-43 for FCD calculations. The range-separation parameter ω was set to 0.14 a.u. and 0.15 a.u. for ωPBE and BNL, respectively, following the scheme previously described.35

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2.3. Electronic coupling. Excited state calculation was performed with TDDFT with B3LYP or LC density functionals, ωPBE and BNL, 6-31G* basis sets and the CRENBL effective core potential.29-32 The electronic coupling between the CT state and the ground state was calculated with the two-state and multi-state FCD scheme36-37. In its common form, the two-state FCD (2sFCD), diabatic states for the reactant (ΨR) and product states (ΨP), is a linear combination of 2 chosen eigenstates that yield the best charge localization, and the coupling (HRP) can be then calculated.37-38, 44 The extent of charge localization is quantified by ∆q, which is defined as ∆$ = %+∈- &'()*( − %+∈0 &'()*(,

(3)

for a state with density ρ(r). In Eq. (3), the integration is carried out with a region for the donor (D) and acceptor (A). An ideal ∆q value for a complete charge-separated state is 2, when the donor carries a positive charge and the acceptor a negative charge. The same quantity can also be defined for the transition density between any 2 states, and thus linear transformation can be performed. In 2sFCD, the diabatic state is the eigenfunction of the 2×2 ∆q matrix of the reactant and product states.37 msFCD involves a series of linear algebraic operations that allows for finding the best charge-separated states within a number of excited states.36 msFCD is useful for cases with large mixing between CT and local excited (LE) states, where the overestimation of the coupling can be corrected. In the TiO2-dye system, a small TiO2 cluster was used in the calculation. Many energetically similar "band-like" molecular orbitals may contribute to a different extent of both CT and LE states. Hence, obtaining an ideal CT state with the 2sFCD approach is difficult. A multi-state treatment helps in reducing the problem of nearly degenerate orbitals by including more eigenstates to determine the best charge-localized states.36-37 msFCD generates better charge-localized states and is more likely to yield reasonable coupling values.36 The definition of donor and acceptor is shown in Chart 2, with the boundary between titania and cyanoacrylic acid. 6 `


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2.4. Natural transition orbitals. Natural transition orbitals (NTOs) are a compact representation of electronic transitions for a given system in the form of pairs of particle and hole orbitals.45-47 For each excited state, the transition is separately transformed in the occupied and virtual orbitals by corresponding orbital transformation. For a transition from the ground state Ψ0 and excited state Ψex, as described by a singleexcitation theory, the transition is described as 123 = 45 78 9:3; :2 95< =,

(4)

where index i is for the occupied orbital, and a is for the virtual orbital. T is a rectangular No×Nv matrix, for a system with No occupied orbitals and Nv virtual orbitals. With unitary transformations U and V, the transit matrix T becomes diagonal, |[? ; 1@]2B | = C2 D2B .

(5)

where U and V are the unitary matrices that diagonalize T†T and TT†, respectively. As a result, the possible number of orbital pairs involved in a transition is reduced from No×Nv to No. Corresponding basis functions are

（F , F , ⋯ , FHI ） = （J , J , ⋯ , JHI ）?,

(6)

and

（F′ , F′ , ⋯ , F′HL ） = （J′ , J′ , ⋯ , J′HL ）@.

(7)

where （J , J , ⋯ , JHI ） and （J′ , J′ , ⋯ , J′HL ） are the occupied and virtual molecular orbitals. （F , F , ⋯ , FHI ） and （F′ , F′ , ⋯ , F′HL ） are the transformed hole and particle NTOs, respectively, with amplitude C2 . We note that the NTO was initially developed for plotting the excitation. In the present work, we used NTO to describe the diabatic states. For each state, we report the particle and hole orbitals with the largest amplitudes.

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2.5. Reorganization energies. The reorganization energy (λ) is a key parameter in the Marcus electron transfer rate theory and describes the effect of external coordinates on the energetics of the reactant and product states. We followed standard approaches previously reported48 describing reorganization energy as: λ = λin + λout .

(8)

The inner-sphere reorganization energy (λin) is calculated as follows: λin = ER(QP) – ER(QR)

(9)

where EA(QB) denotes the energy of state A calculated on the basis of the structure optimized at state B. The calculation of the outer-sphere reorganization energy (λout) is based on the modified Marcus model49 and the PCM solvation model. Detailed procedures for the calculations are included in Supporting Information. The calculated values are also listed in Supporting Information (Table S1). 2.6. Free energy differences. The calculation of the free energy difference is based on the difference between the TiO2 conduction-band Fermi level and the oxidation potential of the dye. The Rehm-Weller equation50 is used to correct for the dielectric solvation energy: ∆G 0 = Eox − Ered − E00 −

q 2  1 1  1 1 1  + − +    4πε 0  2rD 2rA  εref εs  εs dDA 

,

(10)

where Eox and Ered are the oxidation and reduction potentials of the donor and the acceptor; E00 is the zero-to-zero electronic transition energy, which is 0 in this study, because the ground state is involved; rD/A is the radii of the donor/acceptor; dDA is the distance from the center of mass of the acceptor to the semiconductor surface; q is the amount of charge transferred; and ε0, εref and εs are the vacuum permittivity and the dielectric constant for the reference solvent where Eox and Ered are taken and for the solvent of the system where the charge transfer is being modelled, respectively. Here we use the TiO2 Fermi level (-4.3 eV) and the oxditation potential of dye, respectively, for Eox and Ered. THF is the reference solvent 8 `


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(εref = 7.52), and acetonitrile is the solvent for DSC (εs = 36.64). Because a large TiO2 nanoparticle is considered a donor in this case, the term 1/rD is 0. To determine rA, the radius of the spherical cavity of the acceptor (cationic dyes), we calculated the molecular volumes by Jmol51 and determined the radii of spheres with the same volume. All the quantum chemical calculations were performed with a developmental version of QCHEM.52 3. RESULTS AND DISCUSSION For each dye molecule, we first optimized the structure in the cationic state and combined it with a small anatase (TiO2)9 cluster.40-41 We examined 4 models with different TiO2–dye attachments as well as positions of adsorbed proton [e.g., chelating to one Ti atom (chelate), bridging to two Ti atoms (bridge), chelating to one Ti atom with one surface proton (chelateHsurf) and bridging to two Ti atoms with one surface proton (bridge-Hsurf)] (Chart 3). To obtain the CT state and its coupling, we performed TDDFT calculations followed by FCD analyses. We found that a simple 2sFCD cannot always generate proper charge-localized diabatic states, and this problem can be fixed by msFCD. In Table 1, we list the couplings (HRP) and NTOs45 for the diabatic states of FTT as an example. The main configurations (85% ~ 99% in population) of diabatic states from the two-state and multi-state FCD are included. When using two-state calculations with the B3LYP functional, the diabatic state is still a mixture of a local excitation of the dye and a rather small CT fraction, leading to a very small ∆q value (0.097) and a large electronic coupling value (529.3 meV). The multi-state approach offers a much better charge-localized CT state and a large reduction in coupling (29.4 meV), as seen in Table 1. However, the diabatic state derived from msFCD has a very low CT state energy (0.94 eV). The low CT state energy observed is consistent with the known defects of typical GGA density functionals.29-34 In a previous work,35 we also observed low coupling values 9 `



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with GGA density functionals. The wrong asymptotic potentials arising from the traditional functionals affects the quality of the CT states involved and thus electronic coupling. From Table 1, we also see a difference in the results with a LC density functional. With ωPBE, a small delocalization in the electron population is observed with 2sFCD, which is removed when msFCD is used, with the ∆q value increased from 1.82 to 1.95. The coupling value is reduced slightly with msFCD (from 92.7 to 61.1 meV), and the CT state energy is kept at a similar level. Improved quality in the coupling value was reported with LC functionals as compared with a coupled-cluster level of calculation or in experimental results.35 Calculation with range-separation density functionals such as ωPBE avoids the known problems in the CT state, and the msFCD allows for removing the LE component, with an intermediate coupling value we used for further study. In Table 2, we list the energies, difference charges, NTOs of the diabatic states, and the msFCD coupling values for all dye molecules in the 4 models with the ωPBE functional. We found that 15 states were sufficient to remove erroneous LE components and yield CT states with reasonable charge difference values (-1.93 ~ -2.00) in most msFCD calculations. With this approach, we observed well-defined CT states in all cases studied, where the particle and hole NTOs populate at the titania and dye region, respectively. The final msFCD coupling values for the 4 molecules are shown in Figure 2. Calculations with both the cationic and neutral structures of the dye were included; that is, we included electronic coupling for the TiO2-dye system, each with 8 different structural settings. Although we were unable to identify all the structures at the TiO2-dye interface such as other binding modes studied and reported before,53-56 the variation we show to some extent implies that the CRD coupling features a general trend for different systems and is largely independent of the detailed structural model. This weak dependence in the electronic coupling to the detailed structure is a general validation of the Condon approximation.28, 57 10 `


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For each molecule, the coupling values remained largely similar across different binding models, with the smallest coupling being about 1/3 of the largest coupling among the models of the same dye, constituting to, at most, an order of magnitude of uncertainty in the final rates. The coupling values from the neutral structure are mostly larger than those from the cationic structures. Such results are likely from the better charge localization in the cationic structure. Because we restrict ourselves to the cationic structure in the discussion, since they resemble the actual situation for charge recombination more closely, the uncertainty is further reduced. This result also indicates that calculations performed with a cationic structure, a model for the relaxed CT state after charge injection, is an important setting that would affect the resulting coupling. As seen in Figure 3, the choice of density functionals could affect the prediction of couplings. Among the 4 TiO2–dye systems with different binding models, the couplings were generally smaller with the B3LYP hydrid functional than the range-separated functionals such as BNL and ωPBE, similar to previous observations.35 In addition, BNL and ωPBE coupling values are quite consistent with each other. In our previous study,35 coupling values were often slightly larger from BNL than ωPBE in both ground- and excited-state charge transfers. However, here, results from the 2 density functionals are quite similar across the systems we studied. The electronic couplings are about 3 times larger for MZGB1 than MZGB2. MZGB2 has an additional phenyl between the benzothiadiazole and the cyanoacrylic acid anchor. From the NTOs listed in Table 2, the hole populations of the MZGB pair spread from the amine through the thiophene moieties, leaving a small tail residue on the benzothiadiazole group. The additional phenyl group further separate the particle population in titania and the particle population of the dye, for a much reduced coupling.

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With only a difference in the orientation of the thienothiophene group, comparison of the coupling between the 2 TT dyes is rather intriguing. Figure 3 shows a fairly consistent trend with the coupling slightly larger for MCT than FTT. The particle NTO is slightly varied in the lower ring of the thienothiophene moiety. A p orbital of the S atom and the π orbital of the C=C bond in the ring are seen, and their positions are swapped accordingly in MCT and FTT. The π orbital of C=C offers better resonance in the overall π conjugation, and when it is placed near the cyanoacrylic acid and the titania as in MCT, the overall coupling is increased, as seen in Figure 3. This result offers hints for the different performance between the two TT dyes. With the reorganization energy and free energy difference calculated, in Table 3, we summarize important parameters for kET based on Eq. (1). For the two MZGB molecules, the calculated HRP values in the chelate model are 26.8 and 8.9 meV for MZGB1 and MZGB2, respectively (with ωPBE). The nearly three-fold difference is due to the longer D-A separation and the relatively non-planar benzothiadiazole-phenylene spacer for MZGB2. The ∆G0s were mainly determined by experimental redox potentials, and the reorganization energies were from theoretical modeling. The estimated (λ+∆G0) values were -0.42 and -0.31 eV for MZGB1 and MZGB2, respectively, with both in the Marcus inverted region. A faster ET rate would have been expected for MZGB2 according to Eq. (1), if the less-negative ∆G0 value dominated. However, with the inclusion of our calculated electronic coupling, the CRD rate was faster for MZGB1 than MZGB2. The lifetime (1/kET) for MZGB1 (4.99 ps) is calculated to be about 70% shorter than that for MZGB2 (15.84 ps). Compared to the experimental data obtained from ultrafast transient absorption measurements, 18 and 526 ps for MZGB1 and MZGB2, respectively,15 we show good agreement in the relative rates, even though our estimated rates are faster by about 1 order of magnitude. Considering the

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complexity of the system and the first-principle–based methodology without any adjustable empirical parameter in the prediction, the general agreement is still good. To further investigate the structural effects on the coupling, we included the couplings and energies at different twist angles between the aromatic fragment benzothiadiazole (Btd) and its adjacent carboxylic (Ac, for MZGB1) or phenyl (Ph, for MZGB2) group in the optimized cationic MZGB dyes (Figure 4a). The HRP value is decreased with increasing twist angle (Figure 4b). For MZGB2, the energy barrier toward the lower twist angle is rather low, and the coupling value varies slightly in this region. Therefore, if an ensemble of different twist angles in this region were included, the result would have been quite similar. However, the CRD for both FTT and MCT molecules have more negative (λ+∆G0) values (-0.60 and -0.52 eV for FTT and MCT, respectively). The calculated HRP values were 61.1 and 89.3 meV for FTT and MCT, respectively. With more negative free energy change and slightly less electronic coupling in FTT, the overall kET favors MCT (6.63 × 1011 s-1) over FTT (9.84 × 1010 s-1). We previously found that the Mulliken charge difference on excitation is a promising indicator for coupling strength from the dye molecule to the attached TiO2 nanoparticle when a photon is absorbed.58 However, this pair of dyes shows quite similar changes in Mulliken charges, possibly because of the very similar molecular structure.39 The difference in CRD rates may contribute to considerable differences in photocurrent output. With transient photovoltage-decay measurement, the electron lifetimes were 0.53 and 0.44 ns for the iodine-free cells fabricated with FTT-1 and MCT-1, respectively (details in Supporting materials). Without iodine electrolyte, the decay of free electrons in TiO2 may proceed predominantly with CRD. Perhaps with a wide range of different microenvironments for the dye, the nanosecond resolution in the measurement picks up the slower components in the population. Because the fast CRD rate calculated is derived from a set of model structures that are rather regular, the difference in the CRD rates of the 2 systems mainly depends on 13 `



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their redox potentials (∆G0s) and electronic couplings to TiO2. Therefore, there still is a chance for FTT to be slower than MCT in overall CRD rates, even if the reported experimental rates are from dyes in a different microenvironment. The slower CRD rate may be an important key for its better performance, such as significant increment of JSC.39 A comparison of the 2 pairs of TiO2–dye+ systems demonstrates that both ∆G0 and HRP can play a crucial role in the CRD rates. The Marcus rate dependence on ∆G0 for the 4 dyes is included in Figure 5. To show a range from our theoretical prediction, the data in Figure 5 include different coupling values from all 4 different TiO2−dye+ binding modes and the 2 different λout values estimated. The 2 curves for MZGB1 and MZGB2 are well separated, which indicates the dominating influence of the coupling values, regardless of different binding models. However, the CRD rates for MCT are in general comparable to those for FTT, because an overlapped region in the Marcus rates is seen. In this case, the more negative ∆G0 value in FTT leads to a slower CRD rate. Our current computation model did not include a “trapped” local electron that may be from the defect or impurity/dopant in TiO2, which is often used in empirical models for charge recombination kinetics. Exponential energy dependence of the TiO2 state is often assumed,5960

when charge recombination to the electrolyte is studied. The exponential energetic

distribution of trapped electrons offers a good interpretation of recombination kinetics of trapped electrons on TiO2. Without knowing the identity and structure of the trapped states, performing first-principle calculations is difficult. Nevertheless, our work offers a basis for studying charge recombination back to the oxidized dye, which may be extended for the trapped state in the future. On the other hand, we also note that our msFCD approach produces an LUMO state that is localized at TiO2. We expect that the exponential decay character in the donor-acceptor distance, as observed previously,61 is preserved, since it is a general character of the diabatic wavefunctions, as seen in our previous works.36, 62 14 `


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The systems studied in the present work cannot be calculated with a more rigorous theoretical model. However, the use of LR-TDDFT for electronic coupling has been rigorously compared with higher-level theories (for small systems) or with experimental results (for systems with available spectroscopic data).35 Nevertheless, a careful assessment can be made by examining the results, such as the very low CT diabatic state energies (from B3LYP) and the local excitation components in the NTO analysis, etc. Since charge transfer problems with a semiconducting cluster can be prone to many problems in low-cost computational schemes, and with the importance and the need for predicting charge recombination rates, we believe that the strategies we present for avoiding known computational artifacts is important for future applications. In this work, we demonstrate that electronic coupling can be calculated without apparent artifacts, from the selection of the density functional, to the treatment of many nearly degenerate TiO2 orbitals. With DSCs, it is desirable to reduce the CRD rate. Since the reaction is exothermic and usually at the Marcus inverted region, the (λ+∆G0) value should be as negative as possible. This situation can be achieved by reducing λ or increasing the negative ∆G0 in its amplitude. Reducing λin requires careful planning of the molecular design,63-65 whereas reducing ∆G0 would also change the absorption spectra and affect the charge injection efficiencies. A third possibility is to lower the electronic coupling for CR. As discussed in this work and previously,15,

39

the twisted angle among aromatic π moieties helps reduce the electronic

coupling between TiO2 electrons and oxidized dyes, which would lead to a reduced CRD rate and benefit cell performance.

4. CONCLUSIONS In this work, we estimated electronic couplings in the CRD reaction. TDDFT with the LC functionals BNL and ωPBE, with subsequent msFCD calculation, yield reasonable diabatic 15 `



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Page 16 of 32

charge-localized state and electronic couplings. msFCD can handle the nearly degenerated active orbitals involved in the TiO2–dye cluster systems without the artifact of overestimation in coupling. Together with the free energy differences estimated by experimental values and reorganization energies calculated by using standard methods, our predicted ET rates are largely similar to experimental findings. Despite the general dominating roles for ∆G0 and λ in electron transfer rates, electronic coupling-dominated CRD rates are seen for the MZGB dyes. To the best of our knowledge, this is the first work to report reliable ab initio electronic coupling for charge recombination for a TiO2–dye cluster system. Our work provides the foundation and important insights for future development in the theoretical modeling of DSCs.

ASSOCIATED CONTENT §This article is dedicated to the memory of Dr. Bo-Chao Lin (1976-2016); his passion to science, heart-warming smiles, and all the care and love in the friendship with the authors. AUTHOR INFORMATION Corresponding Author *Chao-Ping Hsu: [email protected]. *Jiann T’suen Lin: [email protected] Present Addresses †H.-H. Chou: Department of Chemistry, National Chung Hsing University, Taichung 402, Taiwan (R.O.C.). ‡C.-H. Yang: Department of Chemistry, University of Colorado Denver, Denver, Colorado 80217-3364, United States

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Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS We thank the Ministry of Science and Technology and the Research Program on Nanoscience and Nanotechnology of Academia Sinica for financial support. HHC greatly thanks Dr. ZhiQiang You (Q-Chem, Inc.), Dr. Bo-Chao Lin (Institute of Chemistry, Academia Sinica) and Dr. Chuan-Pei Lee (Department of Chemical Engineering, National Taiwan University) for their kind help. CPH acknowledges the National Center for Theoretical Sciences.

Supporting Information. Additional computational and experimental data are included in the supporting information. This material is available free of charge at http://pubs.acs.org. REFERENCES 1. Grätzel, M., Solar Energy Conversion by Dye-Sensitized Photovoltaic Cells. Inorg. Chem. 2005, 44, 6841-6851. 2. Hagfeldt, A.; Grätzel, M., Molecular Photovoltaics. Acc. Chem. Res. 2000, 33, 269277. 3. Hagfeldt, A.; Boschloo, G.; Sun, L.; Kloo, L.; Pettersson, H., Dye-Sensitized Solar Cells. Chem. Rev. 2010, 110, 6595-6663. 4. Durrant, J. R.; Haque, S. A.; Palomares, E., Towards Optimisation of Electron Transfer Processes in Dye Sensitised Solar Cells. Coord. Chem. Rev. 2004, 248, 1247-1257. 5. Haque, S. A.; Handa, S.; Peter, K.; Palomares, E.; Thelakkat, M.; Durrant, J. R., Supermolecular Control of Charge Transfer in Dye-Sensitized Nanocrystalline TiO2 Films: Towards a Quantitative Structure-Function Relationship. Angew. Chem. Int. Ed. 2005, 44, 5740-5744. 6. Haque, S. A.; Palomares, E.; Cho, B. M.; Green, A. N. M.; Hirata, N.; Klug, D. R.; Durrant, J. R., Charge Separation versus Recombination in Dye-Sensitized Nanocrystalline Solar Cells: the Minimization of Kinetic Redundancy. J. Am. Chem. Soc. 2005, 127, 34563462. 7. Clifford, J. N.; Martinez-Ferrero, E.; Viterisi, A.; Palomares, E., Sensitizer Molecular Structure-Device Efficiency Relationship in Dye Sensitized Solar Cells. Chem. Soc. Rev. 2011, 40, 1635-1646.

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λ 2(HRP)

TiO2--Dye+ (Ψ ΨR)

∆G0

TiO20-Dye0 (Ψ Ψ P)

Figure 1. Free energy curves for a CRD process. The curved solid lines in the inset represent the avoidance of the crossing of 2 adiabatic states and the crossed dotted curves the crossing of diabatic states.

Figure 2. msFCD coupling calculated with the cationic (C) or neutral (N) structures of the dye in the 4 binding models, calculated with ωPBE.

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Figure 3. Calculated msFCD coupling of all TiO2-dye systems with B3LYP, BNL and ωPBE functionals.

23 `



(a)

O (TiO2)9

CN

O

S

O

N

S N

S

N

-2.0o

1.3o

(φBtd-Ac)

2.0o

O

-6.1o

MZGB1 O O

CN

(TiO2)9

S

O

N

S N -4.6o

S

33.3o

N 1.5o

(φBtd-Ph)

-3.4o

O

5.3o

MZGB2 (b) 35

0.6

HRP(MZGB1) HRP(MZGB2)

Relative energy (MZGB1). Relative energy (MZGB2)

25

0.5 0.4

20 0.3 15 0.2 10 0.1

5

Relative Electronic Energy (eV)

30

msFCD Coupling (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 32

0.0

0 10

20

30

40

50

60

70

80

90

Twist Angle (degree)

Figure 4. (a) Dihedral angles between aromatic moieties in the ground cationic state of MZGB1 and MZGB2; (b) calculated msFCD coupling (solid triangles) and relative energies (solid lines with open triangles) with different twist angles, FMNOPQ and FMNOR as defined in (a) for MZGB1 and MZGB2, respectively. Results are from the chelate model.

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Figure 5. CRD rates of different TiO2−oxidized dyes and their dependence on ∆G0. The Marcus parabolas are depicted in color curves, with the upper limit (calculated with the largest coupling and the smaller λout) and lower limit (smallest coupling with the larger λout) depicted for each molecule, and the area between filled with shaded colors. Circles, diamonds, triangles, and squares are for chelate, bridge, chelate-Hsurf and bridge-Hsurf binding models, respectively, as indicated in Figure 2, with closed symbols using λout derived from the PCM solvation model and open symbols from a dielectric continuum model. All data are colorcoded for the 4 dye molecules studied as indicated in the inset.

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Page 26 of 32

Chart 1. Molecules studied in the present work.

TT dyes

MZGB dyes C6H13O

S N N

NC

S

N

S

N

CO2H

NC C6H13O

S S

1

FTT-1

C6H13O

O

O

S N N

NC

S

N

NC C6H13O

S

N

S

CO2H

CO2H

CO2H

S

2

MCT-1 O

O

Chart 2. Donor and acceptor definition of the TiO2–dye cluster.

N O "TiO2"

dye

O

donor (D)

acceptor (A)

Chart 3. Schematic representation of (a) the (TiO2)9 clusters and (b) the binding models for each of the dyes. (a)

(b)

O Ti O H Ti O Ti O O O O O Ti1 Ti Ti OOO O O Ti O 2Ti O O Ti O O chelate

O Ti O Ti O Ti O H O O O O 1 Ti (TiO2)9 Ti O Ti O OO O Ti O 2Ti O O Ti OO H proton from carboxylic acid bridge

N

H

O

(TiO2)9

dye

O

chelate-Hsurf

dye

O

(TiO2)9 H

bridge-Hsurf

N O

H

(TiO2)9

(TiO2)9

O

(TiO2)9 H

N

O

dye

O

N

O H

dye

The bold symbol Ti1 is the single binding site for the chelate model. For bridge models, both Ti1 and Ti2 are connected to the O atoms of the dye. We also studied 2 different sites for the proton originating from carboxylic acid, one denoted as Hsurf (marked in bold in (a)). 26 `


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Table 1. Natural transition orbitals (NTOs) of the CT state in FTT with a chelate structure. Functional

NTOb

FCD

CT state energy, eV

2sFCD

2.15

0.097 529.3

85%

msFCD

0.94

1.983

29.4

99%

2sFCD

2.09

1.823

92.7

98%

msFCD

2.10

1.947

61.1

99%

∆qa

HRP, meV

Hole

Particle

Population

B3LYP

ωPBE

a

∆q is the difference charge in the CT state, defined in Eq. (3). bOrbitals are plotted at isovalue ±0.01 a.u.

27 `



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Page 28 of 32

Table 2. Lowest CT eigenstates, diabatic states, NTOs of the diabatic state, and msFCD couplings for the diabatic CT state with the ground state of dyes based on different simulation models.a System Binding Model

Eigenstate Diabatic CT Coupling, HRP ∆q (meV) energy (eV) energy (eV)

NTOb Hole

Particle

Percentage

chelate

2.44

2.45

-1.99

26.84

>99%

bridge

2.74

2.74

-1.99

13.36

>99%

chelate-Hsurf

2.84

2.84

-1.99

14.53

>99%

bridge-Hsurf

2.93

2.93

-1.99

15.97

>99%

chelate

2.33

2.33

-1.99

8.85

>99%

bridge

2.70

2.72

-1.99

4.00

>99%

chelate-Hsurf

2.71

2.71

-2.00

3.76

>99%

bridge-Hsurf

2.92

2.91

-1.99

3.79

>99%

MZGB1

MZGB2

28 `


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


chelate

2.09

2.10

-1.95

61.06

99%

bridge

2.67

2.67

-1.97

40.53

99%

chelate-Hsurf

2.47

2.43

-1.96

77.46

99%

bridge-Hsurf

2.81

2.81

-1.93

62.03

99%

chelate

2.32

2.27

-1.94

89.25

99%

bridge

2.66

2.65

-1.97

51.06

99%

chelate-Hsurf

2.72

2.70

-1.95

102.78

99%

bridge-Hsurf

2.81

2.80

-1.93

64.75

99%

FTT

MCT

a

Data were calculated based on geometries at the reactant state, with the ωPBE functional. bOrbitals are plotted at isovalue ±0.02 a.u..

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Table 3. Calculated CRD parameters using Eq. (1). MZGB1

MZGB2

FTT

MCT

HRP,a meV

26.84

8.85

61.06

89.25

∆G0,b eV

-1.09

-1.00

-1.31

-1.25

λ,c eV

0.68

0.69

0.72

0.73

(λ+∆G0), eV

-0.42

-0.31

-0.60

-0.52

kET, 10 s

20.0

6.31

9.84

66.3

τ, ps

4.99

15.84

10.16

1.51

10 -1

a

The calculation of the msFCD coupling values is based on the chelate model, with the ωPBE density functional. bThe free energy (∆G0) is determined from Rehm-Weller’s equation by using the conduction band edge of the TiO2 semiconductor as -4.30 V vs SCE. c The outer reorganization energy was calculated with the PCM solvation model.

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Table of Contents Graphic

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First-Principle Determination of Electronic Coupling and Prediction of

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