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Resonance State Method for Electron Injection in Dye Sensitized Solar Cells David Sulzer, and Koji Yasuda J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00364 • Publication Date (Web): 04 Sep 2018 Downloaded from http://pubs.acs.org on September 4, 2018

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Journal of Chemical Theory and Computation

Resonance State Method for Electron Injection in Dye Sensitized Solar Cells David Sulzer† and Koji Yasuda∗,‡,¶ †Institute for Molecular Science, 38 Nishigo-Naka, Myodaiji, Okazaki, 444-8585, Japan

a

‡Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan ¶Institute of Materials and Systems for Sustainability, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan E-mail: [email protected] a

present address: Institute of Physical Chemistry, Theoretical Chemistry Group, Karlsruhe Institute of Technology, Fritz-Haber-Weg 2, 76131 Karlsruhe, Germany, [email protected]

Abstract Herein, the ab initio method is applied to examine metastable molecular excited states on a solid surface using resonance state theory and Green’s function. A formula for the complex energy correction that determines the decay rate is presented; the configuration interaction effect together with major molecule-surface interactions are considered in more detail as compared to previous studies. Furthermore, the lifetimes of the excited states of Ru-terpyridine dyes adsorbed on an anatase surface are calculated, and the effects of the molecular structure and adsorption mode on the electron injection rate are studied. Also, the adsorption structures and relative stabilities of a series of Ruterpyridine dyes - including the black dye - are reported. An implicit solvation model is necessary to reliably calculate the alignment between the photoabsorption spectrum and the conduction band density of states, governing the injection rate. Finally, some of the factors that limit the injection ability of dyes are discussed.

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1

Introduction

Dynamical processes of excited molecules adsorbed on solid surfaces are observed in various fields of science. Organic molecules adsorbed on surfaces are studied for possible applications in molecular 1 or spintronic 2 devices, and their electronic structures are usually examined via spectroscopic methods. Dye-sensitized semiconductors are widely investigated owing to their potential use in solar cells 3 and hydrogen-generating devices. 4–6 Although the initial enthusiasm for dye-sensitized solar cells (DSSCs) is decreasing since the emergence of the new perovskite cells, 7 basic studies on interface electron transfer process are invariably important. 8,9 Systems with continuous energy spectra exhibit qualitatively different behavior as compared to systems with discrete spectra; for example, the photoabsorption intensity over the first ionization potential (IP) shows a characteristic asymmetric shape due to the interference, known as the Fano profile. 10 These highly excited states are metastable and decay to various ionized states. The ab initio method enables the analysis of such phenomena and has been successfully applied to simple systems such as atoms and molecules in vacuum. However, it was only recently that quantum chemists began to study dynamical processes on surfaces, because of the complexity of the systems and the lack of an efficient method. Under proper conditions, a photoexcited molecule on a semiconductor surface injects an electron into the conduction band (CB) during de-excitation, which is observed in DSSCs. Time-dependent quantum simulations of surface models offer a promising tool to elucidate the process. In this method the time-dependent Schrödinger equation of electrons is solved numerically from a given initial wave function, and the wave packet motion into the semiconductor expresses the electron injection. Several notable works on dye-sensitized (dyed) surfaces have been reported thus far. Prezhdo et al. reported molecular dynamics (MD) simulations of a model chromophore with nonadiabatic surface hopping. 11–14 Meng et al. analyzed the injection and recombination dynamics of excited organic dyes. 15 Typically, the electron-injection time is much longer than the timestep of the wave packet propagation. 2


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Owing to the resulting high computational cost, the method was applied to selected dyes with various approximations, such as the use of cluster models and/or fast and approximate ab initio methods. In addition, the preparation of a proper initial wave function is not trivial because there is an infinite number of eigenstates, and the initial wave function uniquely determines the final injected state through the unitary time evolution, and hence, the injection rate. Another theoretical method for electron injection uses Fermi’s golden rule formula of the transition probability to the band continuum. This procedure requires less computation time, and the results are easier to analyze as compared to quantum simulations. Li et al. and Triosi et al. reported notable results based on the ab initio Hamiltonian. 16–20 In this type of method, the interaction between the dye and the surface that causes the transition should be treated as a perturbation. However, as this chemical interaction is not necessarily weak, it is desirable to have a better unperturbed model, which is at least as accurate as the cluster model of the dyed surface. Moreover, theories are often derived from the one-body model Hamiltonian, which is less appropriate for excited molecules. The existence of an infinite number of eigenstates in extended systems is the source of the computational difficulty. However, by applying outgoing boundary conditions to the Schrödinger equation, 21 we can eliminate most of them, except for the most interesting eigenstates that describe the metastable excited states. Based on this idea, we previously proposed a method to calculate the resonance states on a surface. 22 At the lowest order the proposed equation reduces to eq. 7 (see below), which is more detailed than the previous ones 17–20 based on Fermi’s golden rule. In this study, the equation is applied to a famous Ru dye - called black dye (BD) - and its analogs. In addition, we report subtle, but important, issues related to actual ab initio calculations: (i) Because of the surface dipole we have two vacuum levels, and the calculated orbital energies should be aligned to a common reference. (ii) Without a proper solvent model, tiny structural modulations may induce sizable changes in the surface dipole and orbital energy.

3



These two issues are common to many ab initio studies of molecular adsorbed surfaces. The remaining paper is organized as follows: Section 2 summarizes the resonance state method; section 4 reports the stable adsorption structures of a series of Ru-terpyridine dyes on anatase (101) surfaces under experimentally supported assumptions; section 5 presents an analysis of the energies of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) for various dyes and adsorption modes. We analyze the excited states of the dyed surface with the cluster model in section 6 and the band structure and self-energy (SE) of the anatase (101) surface in section 7. Considering all factors, we present the results of the electron injection rate in section 8.

2

Summary of the resonance state method

In a previous paper, based on the resonance wave function, we reported a rigorous method for studying electron injection processes by first principles calculations. 22 We modeled the process as the autoionization of the superexcited state which lies above the first IP. 10 In contrast to the usual stationary wave function the resonance wave function is characterized by an asymptotically divergent spherical wave, 21,23,24 and because of this, the energy becomes complex, E = E0 − iΓ/2, with the imaginary part giving the state lifetime Γ. The complex coordinate method, 25 in which the radial coordinate r of an electron is replaced by the complex number r exp(iθ), and the method of the complex absorbing potential, 26 which places the complex absorbing potential on the asymptotic region of the wave function, were successfully applied to atoms and molecules in vacuum. 27 However, we think that none of them is suitable to solve our problem having complicated boundary conditions. To express the boundary conditions, we adopted the nonequilibrium Green’s function (GF) method, which is used in theoretical simulations of molecular devices. In the simulations of a molecular device, a molecule connected to two leads, one-particle GF is calculated under the boundary condition that an orbital becomes the right-going asymptotic wave in

4


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the leads. If there is one lead, the pole of the GF in the complex energy plane gives the quasi-particle energy, whose imaginary part determines the decay rate of the metastable ion. Though GF does not give the excitation energy, we adopted the idea to express the effect of leads, in our case the bulk anatase, by SE σ(ω), which is the frequency (ω) dependent complex potential. Solution of the Schrödinger, time-dependent density functional theory (TDDFT), 28 or Bethe-Salpeter equation (BS) 29 with this potential describes the resonance state. We think that solving the BS is the most transparent method, although there are several studies on TDDFT descriptions of resonance states. 30 For simplicity, we calculated the zeroth order wave function by TDDFT, which can be regarded as the accurate approximation to the BS, 31 and the effect of SE was perturbatively calculated using eq. 7 (see below). A typical lifetime of 102 femto seconds in a DSSC system corresponds to an imaginary part of Γ = 10−4 atomic units. The perturbative correction to the imaginary part of the energy can be reproduced from Fermi’s golden rule under a stronger orthogonality assumption than that previously made. 22 The transition probability from the initial state I to the final state F under constant perturbation V is, 32 dw =

2π |VF I |2 δ(EF − EI )dν h ¯

(1)

where VF I is the matrix element between the states. The final states may be degenerated and eq. 1 gives the transition probability to one of them. Dirac’s δ-function ensures the energy conservation. We express the initial state energy as EI = ωsinglet + E0 , where ωsinglet is the excitation energy and E0 is the ground-state energy. The final state energy is assumed to be EF = −ϵj + ϵk + E0 , where ϵj and ϵk represent the oxidation potential and the electron attachment energy, respectively. The electron-hole interaction in the final state is assumed to be zero. Because of energy conservation, injection takes place only at the state with ϵk = ωsinglet + ϵj . In DSSC systems ϵj is the lowest oxidation potential of the adsorbed dye and is close to the HOMO energy, ϵH . If we apply DFT with the exact functional they are the same. 33 5



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On the other hand, ϵk is more difficult to calculate. DFT cannot yield exact values except for the lowest one, and that requires an exact and nonanalytical functional. 33 The GW approximation 31 of GF is a reliable method, but because of the computational cost, orbital energies with carefully selected functionals are often used. Similarly, the degeneracy of the final states ρ(ϵk ) is approximated as the CB density of states (DOS). Assuming that the VF I values are the same for degenerate final states the transition probability per unit energy area is dw =

2π |VF I |2 ρ(ωsinglet + ϵH ) h ¯

(2)

which corresponds to eq. (2.60) of Ref. 34. We can derive the detailed equation when the wave functions are the product form. The initial wave function is assumed to be the direct product ΨI = Ψsinglet Φgs of the semiconductor ground state Φgs , and the following excited molecule,

Ψsinglet =

∑ j,b

] 1 [ Ψbj √ a†bα ajα + a†bβ ajβ Ψgs , 2

(3)

where Ψgs is the ground state of the molecule, Ψbj is the configuration interaction coefficient, and a†j and aj are the creation and annihilation operators, respectively. Ψsinglet is the oneelectron-one-hole excited state of the random phase approximation. 35 Similarly, assuming that the excited electron and hole in the final wave function are located at the kth CB orbital and the jth MO, respectively, we have ] 1 [ † † √ ΨF = akα ajα + akβ ajβ Ψgs Φgs . 2

(4)

If the perturbation is a spin-independent one-body potential and Ψgs Φgs is a Slater determinant, after simple calculation eq. 1 reduces to ∑ k

h ¯ dw = −2

∑

[ ] ′ ′ ΨbH ΨbH Im σ(ω + ϵH )bb ,

b,b′

6


(5)

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[ ] ∑ ∑ (k) (k) ′ (b′ ) ′ λ′ Im σ(ω)bb = −π c(b) c V V cλ cλ′ δ(ω − ϵk ), ′ µλ µ µ µ µµ′ λλ′

(6)

k (b)

where Im[σ] is the imaginary part of SE, cµ is the bth MO coefficient, and Vµλ is the perturbation in the atomic orbital (AO) basis. This is consistent to the state lifetime Γ = −2Im(∆E), where at the lowest order 22 ∆E ≈

vir ∑ occ ∑ α,λ

3

Ψαβ Ψλβ σ(ϵβ + ω)λα .

(7)

β

Computational conditions

The unit cell for the structure optimization and band-structure calculation consisted of a dye molecule and two layers of (TiO2 )24 , which is consistent with the experimental dye coverage. 36 This unit cell was extracted from an anatase crystal with the surface defined along the (101) direction. It contains twelve Ti5c atoms on each surface. The slab was repeated periodically every 30 ˚ A along the direction normal to the surface. During the geometry optimization, the second (TiO2 )24 layer was fixed, and the counter-ion of the dye (if present) was replaced by a homogeneous background charge. We used the Vienna ab initio simulation package (VASP) 5.3.2 37–39 with the projector augmented-wave PBE potential, 40,41 and 3 × 3 × 1 Monkhorst-Pack k-points for Brillouin zone sampling. The PBE 42 functional was used to optimize the structure, while the B3LYP functional, 43 which correctly reproduces the orbital energies of both Ru dyes and anatase, was used to examine the band structure. We did not use long-range corrected functionals such as LC-PBE, because they overestimate the anatase band gap. Similarly, it was reported that MPW1K and CAM-B3LYP failed to reproduce the band gap. 44 The optimization was stopped when the force acting on an atom was below 0.05 eV/˚ A. Solvent effects were taken into account using the polarizable continuum model (PCM) and the linearized Poisson-Boltzmann (PB) model, 45 which assume that the relative permittivity

7



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is determined by the electronic charge density of the solute through the following equations:

ϵ(nsolute (r)) = 1 + (ϵb − 1)S(nsolute (r)), 1 S(nsolute (r)) = erfc 2

{

log(nsolute /nc ) √ 2σ

(8)

} ,

(9)

where ϵb is the relative permittivity of acetonitrile (CH3 CN, 35.688). The parameter nc determines at what value of the electron density the dielectric cavity forms, and the parameter σ determines the width of the diffuse cavity. Since these two parameters are not known for acetonitrile, we determined them for eq. 8 to reproduce the solvation energy of the PCM 46 using GAUSSIAN09. 47 Figure 1 compares the solvation energies of small organic molecules using the parameters nc = 10−3 and σ = 0.6. The reference GAUSSIAN09 calculation used the default settings, that is, the integral equation formalism variant of the PCM and the atomic radii from the universal force field scaled by 1.1 as cavity spheres. As shown in Figure 1, the above parameters reproduce the reference solvation energies well. We used these PCM parameters throughout this paper. The PB model was used to represent the electrolyte, in A. This value corresponds to the which the Debye length (LAMBDA D K) was set to 2.451 ˚ acetonitrile solution of the 0.7 M monovalent salt. In addition to the implicit solvent, we added nine acetonitrile molecules to the topmost Ti5c atoms in the unit cell, to represent the ordered monolayer observed in the previous MD simulations. 48–50 There are three empty Ti5c sites for dye coordination. We used the cluster model for the TDDFT calculations, following the procedure described in Ref. 22. The TiO2 bottom layer in the optimized dyed slab model was eliminated, and only the oxygen positions on the cluster edge were relaxed by the DFT calculations. In the cluster and periodic DFT calculations using GAUSSIAN09, the 6-31G basis set 51–53 was used, except for the Ti, Ru, and S atoms. For Ti, the most diffuse sp function was removed 54 from the 6-31G basis set to avoid numerical instabilities in the periodic boundary condition (PBC) calculations. To avoid artificial polarization we used the same quality, double-ζ basis

8


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set for dyes. For Ru and S, the LANL2DZ basis set was used. 55–57 The basis set with the B3LYP functional successfully reproduced the photoabsorption spectra of the dyes in the visible region obtained in previous theoretical and experimental studies. For example, the excitation energies of three main peaks of the BD below 2.5 eV matched those by the triple-ζ polarization basis within about 0.05 eV. Unless otherwise noted, all the cluster DFT calculations were performed using the B3LYP functional 43 and the GAUSSIAN09 program. To model the clean surface the adsorption structure of CH3 CN on a four-layer anatase (101) slab was calculated by the same optimization procedure. Acetonitrile molecules are adsorbed on both sides with 100 % coverage. The Fock and overlap matrices were calculated using the GAUSSIAN09 program, the B3LYP functional, PBC, and 32×46 Monkhorst-Pack k-points. The same basis set as in the cluster calculations was used. From these two matrices the surface GF and the SE were calculated as follows. 22,58–60 For a given complex frequency ω, we made a matrix M = ωS − F , where F and S are the Fock and overlap matrices of the second principal layer, respectively. The corresponding transfer matrix between principal layers is denoted as V = ωS1 − F1 , where F1 and S1 are the Fock and overlap matrices between the second and third principal layers, respectively. Then we solve the eigenequation, 

 † −1

 M (V )  (V † )−1







−V   Z1   Z1  =  W,  Z2 0 Z2

(10)

and select the eigenvectors (Z1 , Z2 ), whose eigenvalues satisfy |W | > 1. The surface GF G and SE σ(ω) are given by G = Z2 Z1−1 ,

(11)

σ(ω) = V † G(ω)V.

(12)

If necessary, we regularized singular matrices according to Refs. 58–60.

9



Figure 1: Comparison of the PCM solvation energies in acetonitrile with different implementations (kJ/mol).

4

Adsorption structures of Ru dyes

Ab initio calculations require accurate structures of adsorbed molecules on semiconductor surfaces. In this section, we systematically investigate the adsorption structures of BD and its analogs, which differ in the ancillary ligands (Figure 2) on the anatase (101) surface. We found that the adsorption energies and structures of these dyes were very similar, which implies that our results may be applicable to other metal complexes with the same dicarboxy terpyridine (dctpy) ligand. We solved certain discrepancies from previous studies regarding the most stable structure of BD. The position of the protons was found to be sensitive to the computational condition, which may shift the orbital energy and change the injection rate. The structures of Ru N3 dyes on anatase nanoparticles were intensively studied, and it was found that they are bound to the surface via multiple carboxy groups through bidentate bridging, 61–63 bidentate chelating, or monodentate modes. 64,65 Theoretical studies on N3 dyes demonstrated that multiply anchored structures were more stable than singly anchored

10


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Figure 2: Structure of the black dye analogs. ones. 66–70 On the other hand, the theoretical results obtained for the N749 dye, [(C4 H9 )4 N] [Ru(H3 tctpy)(NCS)3 ] (H3 tctpy = 4,4′ ,4′′ - tricarboxy- 2,2′ :6′ ,2′′ - terpyridine), known as BD, are somewhat confusing. Sodeyama et al. reported that the structure anchored with a protonated carboxy group was the most stable one in vacuum, 71 and that it had almost the same energy as the doubly anchored structure in solution. 49 On the other hand, de Angelis et al. reported stable adsorption structures of the BD dye with two anchoring groups, 72–74 determined by the cluster model and the conductor-like polarization model of solvents. Unfortunately, due to deformations of the cluster model from a perfect surface and the lack of published coordinates, it is not easy to compare the results obtained by different groups. Through a systematic study, we compare the stabilities of various adsorption structures of BD on a perfect anatase surface, solve these discrepancies, and propose a more stable, singly anchored structure than that previously reported. 71 The coordinates of the stable adsorption structures are provided in the supporting information to ensure the reproducibility of the study. 11



4.1

Classification of structures

Figure 3 shows the anatase (101) surface, where Ti5c atoms are potential anchoring points. The anchoring carboxy group(s) may be on the middle (4′ ) and/or on the terminal (4 or 4′′ ) pyridine rings of the terpyridine ligands. They attach on the surface Ti5c atoms in either monodentate (denoted as m) or bidentate (denoted as b) form. We denote the singly anchored monodentate structure as S(m) and the singly anchored bidentate-bridging structure as S(b). The bidentate chelating structure is known to be less stable, 75 so we did not consider it. A proton on the monodentate anchoring group may dissociate and move to one of the O2c atoms on the surface. It can form a hydrogen bond (HB) with the carboxylate group, which provides additional stability. Thus, according to a previous study, 50 we located the proton on the most suitable O2c atom for hydrogen bonding, namely: (i) the proton is on O2c indicated by a black arrow in Figure 3, when a carboxylate group bonds to TiB and TiC . (ii) The proton bonds to the O3c atom between two anchoring Ti atoms from below, as shown in the same figure. The distance between two anchoring oxygen atoms (denoted as O and O′ ) of the desorbed dye is 6.4-6.8 ˚ A, whereas that of the adsorbed N3 dye ranges from 4.9 to 7.4 ˚ A. 70 Thus, we selected three pairs of anchoring Ti5c atoms on the surface, X–A, X–B, and X–C in Figure 3. Due to the stiffness of the terpyridine ligand, only one of the two anchoring carboxy groups can adopt a bidentate-bridging structure, resulting in five possibilities, namely, A(mm), A(bm), B(mm), C(mm), and C(bm). In the A(mm) structure, two carboxy groups are attached to TiX and TiA in the monodentate mode. All the structures of A(bm), B(bm) and C(bm), in which one of the anchoring groups adopted a bidentate-bridging form were unstable because of the strain, except for the one in which the bidentate-bridging carboxylate binds to B and C, and the monodentate binds to X. We denote this structure as C(bm). The same position of the dissociated proton on the surface was chosen as the singly anchored structure. The initial structures for ab initio optimization were generated by attaching the anchoring 12


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groups of the dyes to the anchoring Ti5c atoms in all possible ways. The subsequent optimization yielded about thirty distinct structures. Figure 4 displays the most stable structure of each group as well as other interesting structures, while Table 1 compares the energies and bond lengths of these structures. The most stable structure we found was C(mm), whose dye adsorption energy was 108.4 kJ/mol. The doubly anchored structure of B(mm) is less stable by 31.6 kJ/mol. As shown in Figure 4, the O–O′ distances in the structures A(mm) and B(mm) are 7.14 and 7.24 ˚ A, respectively. These values are comparable to those of previous structures (p2x, d2x and d2). 49,71 By comparing our optimized structures with the figures in the literature, we deduce that the B(mm) structure corresponds to these doubly anchored structures.

Ti5c

C

5.35

X 7.57

O3c 7.57 O2c B A Figure 3: Anchoring points on the anatase (101) surface. The structure C(mm) has a rather short O–O′ distance of 5.89 ˚ A, forced by the distance between two Ti5c atoms (5.35 ˚ A in Figure 3). The energy of the desorbed dye was calculated to elucidate the dye strain energy in B(mm) or C(mm). The dye structure was fixed as the adsorbed one except for two dissociated protons, whose positions were optimized. We found that the dye in B(mm) was 129.7 kJ/mol less stable than the freely optimized structure, while that in C(mm) was 103.8 kJ/mol less stable. Thus, the BD in the C(mm) structure is subjected to a smaller strain, which explains the relative stability between B(mm) and 13



1.48

6.47 2.15

1.55

7.24

2.01

1.78

1.63

1.63 2.09

2.28

2.00 1.63

5.89

2.17 1.37

2.17 7.14 1.42

c: A(mm), 20.6 kJ

2.03 1.35

2.13

6.12

2.00 1.49

1.93

e: C(mm), 0.0 kJ

d: B(mm), 31.6 kJ

2.09 2.08

1.68

b: S(b), 82.4 kJ

a: S(m), 27.7 kJ

1.88

1.68

Page 14 of 55

6.10

f: C(bm), 18.2 kJ

2.00 1.39

g: C(bm)H+ sub , 36.6 kJ Figure 4: Adsorption structure of BD on anatase (101) surface, relative energy (kJ/mol), and selected atomic distances (˚ A). Black: O–O′ , green: Ti–O, blue: hydrogen bonded O–H.

14


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Table 1: Relative Energies (kJ/mol) and Selected Bond Lengths (˚ A) of BD Adsorbed on an Anatase (101) Surface. Length ∆E r(O–O)

A(mm) B(mm) C(mm) C(bm) C(bm)H+ sub 20.63 31.64 0.00 18.17 36.62 7.14 7.24 5.89 6.12 6.10 6.48 6.43 6.90 6.97 6.87 2.17 1.93 2.00 2.00 2.08 2.17 2.01 2.03 2.13 2.09 2.13 2.00 a 1.62 1.37 1.78 1.35 1.49 1.39 1.68a 1.42 1.88 1.63 a Hydrogen bond length toward neighbor dye.

S(m) S(b) 27.72 82.42 6.47 6.44 6.69 7.00 r(O–Ti) 2.15 2.09 2.28 r(O–H)

1.48 1.55

Desorbed 108.38 6.38 6.75 -

C(mm). Another factor that influences the stability is the formation of HBs. The B(mm) structure has weak HBs as manifested by the longer HB distances (of 1.78 and 1.88 ˚ A) as compared to the typical lengths (1.35-1.50 ˚ A). Note, however, that the p2a and p2b structures have longer HBs of 2.07-2.21 ˚ A. 71 De Angelis et al. reported doubly anchored structures on an anatase cluster model (c, c’, d, and d’ in Figure S3 of Ref. 74). These structures seem to correspond to either B(mm) or C(mm), but because of the distortion of the cluster from the perfect surface and the lack of detailed structural data, we cannot establish a clear relation. In these structures both protons are dissociated and only one carboxy group forms a long, weak HB (1.71-1.75 ˚ A). Among various doubly anchored A-type structures, A(mm) [shown in Figure 4(c)] was found to be the most stable (20.6 kJ/mol). This structure has twisted carboxy groups about the terpyridine ring, in which the dihedral angles are 27 and 36 degrees. This deformation is caused by a mismatch between the optimal O–O′ distance in the desorbed BD and the distance of the anchoring Ti5c atoms. Because of the strain the A(mm) structure has Ti–O bonds that are longer (2.17 ˚ A) than the typical bonds measured without strain for ˚). monodentate systems (1.93-2.03 A The C(bm) structure shown in Figure 4(f), in which one carboxylate adopts the bidentatebridging mode, has an energy of 18.2 kJ/mol, which is lower than that of B(mm). A monodentate carboxylate forms a HB (1.49 ˚ A) with a surface OH group. This structure seems 15



to be an intermediate of the conversion from B(mm) to C(mm), except for the surface H+ position. Subsequent proton intercalation results in the new structure, C(bm)H+ sub [shown in Figure 4(g)], which has an energy of 36.6 kJ/mol. The proton bonds to the O3c atom from below to form an OH bond of usual length (0.98 ˚ A). At the same time, the bidentate O–Ti bonds become shorter, decreasing from 2.13 to 2.09 ˚ A. The activation energy of the proton intercalation process was estimated to be 71 kJ/mol for acetic acid. 50 Considering the relative stabilities, we can conclude that this structure is less important than A(mm), C(mm), and C(bm). The relative energies of the singly anchored structures, S(m) and S(b), were calculated to be 27.7 and 82.4 kJ/mol, respectively. We can explain the stability of S(m) in terms of HB formation. As shown in Figure 4(a), this singly anchored structure has an additional HB between the second carboxy group and the surface O2c species. Rotating the second CO2 H group by 180 degrees and breaking this HB results in the same structure as p1b and p1 in Refs. 49,71. Hence, the present singly anchored monodentate structure, having an energy of 27.7 kJ/mol, is more stable than the previous one due to this additional HB. Note that a long Ti–O distance of 2.15 ˚ A was found in S(m). The S(b) structure contains four HBs with the neighbor dyes, making a linear chain along the [010] direction. Nevertheless, this chain structure is less stable than others and may not play an important role in the actual device. The position of the proton on the anchoring carboxy group requires further analysis. Previous calculations in vacuum concluded that the protons bind to the carboxylate groups in the doubly anchored structures (p2a and p2b). 71 Later DFT-MD simulations in an acetonitrile solution predicted that the protons dissociate from the carboxy groups and bind to the surface O2c atoms. 49 Cluster model studies by de Angelis et al. supported the doubly anchored, deprotonated structures. 72–74 We compared the energies of the doubly anchored structures in which the positions of the carboxy protons are different and found that the structure with a surface H–O2c group was slightly more stable when we modeled the ace-

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tonitrile solvent as a monolayer comprsing nine CH3 CN units. The order was reversed when the PCM was used for solvation. The proton position has a small influence on the energy, typically 3-6 kJ/mol per OH bond, except for the B(mm) structure with the acetonitrile monolayer. In this case, the structure with two surface H–O2c bonds was computed to be more stable (by 57.7 kJ/mol). Thus, in this section, we reported the results of the doubly anchored structures with surface H–O2c groups. Our results show that the relative acidity between H–O2c and the carboxylic acid depends on the computational setup. Table 2: Relative Energies (kJ/mol) of the BD Analogs Adsorbed on an Anatase (101) Surface. Dye PRT12 PPY ACAC TF2 T4 AZOL

4.2

S(m) A(mm) B(mm) C(mm) C(bm) C(bm)H+ sub 31.65 25.33 27.16 0.00 23.66 38.00 28.46 23.13 24.92 0.00 19.12 31.16 32.80 25.45 33.40 0.00 20.29 36.69 30.69 24.99 33.89 0.00 20.18 35.80 14.37 32.80 15.82 0.00 2.38 16.71 19.90 23.43 18.10 0.00 7.49 21.63

Black dye analogs

We examined the adsorption structures of six BD analogs on the same surface: PRT12 = [Ru (H2 dcttpy)(NCS)3 ]− , (H2 dcttpy = 4,4′ - dicarboxy- 5′′ -(5-hexyl-2-thienyl)- 2,2′ :6′ ,2′′ terpyridine), 76,77 PPY = Ru (H3 tctpy) (ppy) (NCS), (ppy = 2-phenylpyridine), 78,79 ACAC = Ru (H3 tctpy) (tfpbd) (NCS), (tfpbd = 4,4,4-trifluoro -1-(phenyl) -butane -1,3-dione), 80 TF2 = Ru (H3 tctpy) (bpp), (bpp = 4-(5-hexyl-2-thienyl)- 2,6-bis (5-(trifluoromethyl) -1H-pyrazol -3-yl) pyridine), 81 T4 = [Ru (H3 tctpy) (dpb)]+ , (Hdpb = 1,3-di (2-pyridyl) benzene), 82 and AZOL = [Ru (H3 tctpy) (btab)]+ , (Hbtab = 1,3-bis (1,2,3-triazol-4-yl) benzene). 83 These analogs share the anchoring dctpy substructure, and only the ancillary ligands are different as seen in Figure 2. A hexyl group in PRT12 or TF2 is replaced with a methyl group for simplicity. BD in the lowest fifteen optimized structures was replaced with each dye to prepare the initial structure. The subsequent geometry optimization was done using the 17



same procedure as the one detailed for BD. We found that the optimized structures of these dyes with the same adsorption mode were very similar, which implies that other metal - dctpy complexes are adsorbed on the surface with the same structure. The most stable structures found were C(mm), although it is very difficult to prove that they are the true global minimum. The relative energies of the studied structures with respect to that of the most stable one, C(mm), are summarized in Table 2. The stabilities of all the structures are very similar to those of the BD for the anion PRT12 and the neutral PPY, ACAC, and TF2 dyes, but slightly different for the cation dyes, T4 and AZOL. C(bm) is almost degenerate to C(mm), implying that these two structures coexist. The relative energies of S(m), B(mm), and C(bm)H+ sub are reduced to about 17 kJ/mol, which indicates that the ancillary ligand indirectly influences the adsorption structure via the total charge of the dye.

5

Orbital energy analysis of the adsorbed dye

The decay rate of the metastable excited state given by eq. 7 depends on the MO energies and the CB DOS. In this section we report on the results of the calculations as well as on possible pitfalls of the analysis. Firstly, under PBC, we cannot compare the orbital energies in different calculations when the dyed surface has a nonzero dipole moment, because the electrostatic potential jumps across the surface dipole layer, and the two vacuum levels at z → ∞ and z → −∞ are different. 84 We must clarify the zero potential for each calculation; for example, GAUSSIAN09 in two-dimensional PBC uses their average as the origin of the electrostatic potential. The computation of the surface dipole itself is not trivial with three dimensional PBC, 85 and the zero potential is arbitrary. Clearly a comparison of the MO energies from different calculations is meaningless unless we align them with respect to the common physical reference energy. Since one of the anatase surfaces (z > 0) is covered with the electrolyte in the actual

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system, the vacuum level at z → −∞ provides a transparent physical reference energy. Experimentally, the energy levels are often aligned with respect to the normal hydrogen electrode (NHE), and a computational model of the NHE 86 may provide an alternate reference energy. We found another useful reference energy for theoretical calculations of solvated anatase surfaces, namely, the half maximum point of the CB DOS, which is located at almost the same position for different dyes and adsorption modes. In particular, if the DOS is calculated with the Gaussian smearing of 0.1 eV width, this point appeares at −3.6 eV with respect to the z → −∞ vacuum level. This is a convenient reference point, because DOS in this region is not sensitive to the adsorbed dye, nor to the solvation model, in contrast to the bottom of the CB. The total DOS of a dyed surface calculated without the implicit solvent and aligned with respect to the z → −∞ vacuum level is shown in Figure 5.

Figure 5: Comparison of the total DOS of dyed slab models. The inset shows a magnification near the bottom of the CB. Another difficulty we found was that without an implicit solvent model, tiny structural modulations can induce sizable changes in the surface dipole and orbital energies with respect to the z → −∞ vacuum. For example, the BD anion carries a tetrabutylammonium cation, 19



which was modeled as N(CH3 )+ 4 . We found a few stable isomers, in which only the position of the cation was different, and compared the HOMO energies of these isomers with respect to the half-maximum point of the DOS. As shown in Table 3, they are very sensitive to the structure if we only include monolayer CH3 CN as the solvent: the HOMO energy changes by about 0.6 eV as the cation moves. This also happened for the PPY dye adsorbed with C(mm) mode. There are several stable isomers in which the ancillary ligands of NCS and 2phenylpyridine are interchanged. Their properties are expected to be similar, but the HOMO energies calculated using the B3LYP functional and monolayer CH3 CN differed significantly (by 0.7 eV). Considering that the ab initio optimized structure contains small errors, we cannot have a useful orbital energy to predict the injection rate. Fortunately, this problem can be solved by considering the implicit solvation of PCM or PB, as shown in the Table 3: the differences were reduced to 0.03 eV. The periodic surface model represents a perfectly ordered specific isomer, but this ideal structure is seldom realized in the actual system due to fluctuations. Thus, we conclude that the large change in orbital energy can be attributed to an artificial amplification of the unscreened electrostatic effect of the same, repeated structure in the PBC model. Note that theoretical studies of the injection rate of DSSCs without a proper solvation model inherently suffer from ambiguities in the CB position, so one has to manually choose this very important parameter. In this paper, the orbital energies are calculated with the B3LYP functional and the PB solvation model and aligned with respect to the half maximum point of the CB DOS, unless otherwise noted. Table 3: HOMO Energies of Structural Isomers (eV). All the Methods Contain Nine CH3 CN Molecules as the Solvent. Without Implicit Solvation, Small Structural Changes Can Induce Large Fluctuations. Molecule BD+N(CH3 )+ PPY 4 Functional PBE PBE B3LYP B3LYP PBE B3LYP Solvent vacuum PCM PB vacuum PB PB Isomer A -1.87 -1.42 -2.50 -3.25 -1.42 -2.50 Isomer B -1.28 -1.46 -2.50 -2.54 -1.39 -2.60

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Below we report the HOMO/LUMO energies of representative dyes adsorbed on an anatase surface in the S(m), A(mm), C(mm), and C(bm) modes. These dyes have an approximate C2v point group symmetry and the same terpyridine ligand. The LUMO and next LUMO of the desorbed dye are the π ∗ orbitals of the terpyridine ligand and are close in energy. The HOMO and next HOMO are the bonding orbitals between Ru(d) and the ancillary ligands. The HOMOs of T4 and AZOL and the next HOMO of TF2 have symmetries that are different to those of other dyes, and as a result these dyes exhibit dipole-forbidden transitions in the low-energy region. Dyes containing NCS ligands have higher HOMO energies than others. The HOMO or next HOMO have small amplitudes on the anchoring carboxy group. Table 4: HOMO Energy of a Dye in Various Adsorption Modes (Shift from Desorbed in eV). Structure Desorbed S(m) A(mm) C(mm) C(bm) Average

BD PRT12 PPY ACAC TF2 T4 AZOL Average Average (cluster) -5.31 -5.23 -5.28 -5.54 -5.63 -5.59 -5.55 a a a -0.85 -0.90 -0.82 -0.88 -0.86 0.25 -0.66 -0.66 -0.69 -0.64 -0.71 -0.74 -0.68 -0.68 0.32 -0.75 -0.75 -0.80 -0.72 -0.78 -0.81 -0.76 -0.76 0.26 -0.74 -0.81 -0.85 -0.78 -0.83 -0.86 -0.82 -0.81 0.27 -0.75 -0.73 -0.81 -0.74 -0.80 -0.80 -0.75 -0.77 0.28 a not calculated.

In Table 4, we report the HOMO energy of the dyed slab model in PB solvation with respect to the z → −∞ vacuum. The first row shows the value of the desorbed dye in the PCM while the other rows show differences from it. When a dye is adsorbed on the surface with two anchoring carboxy groups in a mode, there is another stable structure, which is obtained by exchanging the anchor groups. In this table, we report the average of HOMO energies of these two structures. From the table we can see that each adsorption mode changes the energy of the respective dye by a similar amount; for example, among the four adsorption modes, the HOMO in the S(m) structure is the deepest for all the dyes, while that of the A(mm) structure is the shallowest. As for the injection rate, since the CB DOS at the HOMO energy plus the photon energy matters, the A(mm) structure benefits from the highest DOS, and the most stable structure of C(mm) follows. The HOMO energies for 21



different adsorption modes distribute within ±0.1 eV around the average value shown in the last row. To examine the model dependence we analyzed the HOMO energies of the adsorbed dyes on the cluster model of Figure 6(a) using the same basis set and the PCM solvation model. As seen in Table 4, the energy increased by about 1.05 eV, which can be attributed to differences in the solvation model and surface dipole effects. The relative order among adsorption modes was the same, in average, but the differences were underestimated. Table 5: LUMO Energy of the Adsorbed Dyes (eV). Structure Desorbed S(m) A(mm) C(mm) C(bm) Average

BD PRT12 PPY -3.14 -2.99 -3.00 a -0.83 -0.83 -0.60 -0.53 -0.58 -0.73 -0.72 -0.71 -0.68 -0.71 -0.76 -0.71 -0.65 -0.72 a not

ACAC TF2 -3.22 -3.12 -0.79 -0.82 -0.57 -0.62 -0.68 -0.71 -0.71 -0.75 -0.68 -0.73 calculated.

T4 AZOL Average -3.14 -3.11 a a -0.82 -0.62 -0.59 -0.58 -0.72 -0.69 -0.71 -0.75 -0.73 -0.73 -0.70 -0.67 -0.69

Since the LUMO energy of a dye is often used to predict the electron injection ability, we analyzed this parameter. We identified the energies of the virtual orbitals of the adsorbed dye as the local maximum points of the dye projected DOS (PDOS) with a Gaussian width of 0.05 eV. Table 5 summarizes the peak position shifts of the PDOS from the desorbed dye for the given adsorption modes. The relative positions are the same as those for the HOMO: S(m) < C(bm) ≈ C(mm) < A(mm). Thus, judging from the CB DOS, the injection becomes more likely in the order described above. The shift values are smaller than those for the HOMO, in average, but the variances among the adsorption modes are larger, which is possibly due to the different interaction with the CB or the different surface dipole potential at the LUMO.

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6

Analysis of the excited states of the cluster model

The present resonance state method estimates the decay rate of the unperturbed excited state using eq. 7. In this section, we examine the nature of the unperturbed states around 1.5 eV, and the effect of the adsorption mode on it. Because of the computational cost, we calculate the excited states by TDDFT using the B3LYP functional and the cluster model. Our model, shown in Figure 6(a), retains the positions of the atoms except for the oxygen atoms on the cutting surface, and at the same time, it is free from artificial excited states due to dangling bonds. 22 We also examine another cluster model [Figure 6(b)] to see the model dependence. Generally, a Ru dye has a metal to ligand charge transfer type excitation in the visible light region; for example, the excitations of the BD below 3 eV are the transitions from the bonding orbitals between Ru(d) and the NCS ligands to the π ∗ orbitals of the terpyridine ligand, with oscillator strengths of 0.03-0.05 atomic units. Transitions from the dye to the anatase CB also appeared with vanishing oscillator strengths. We report the position of the first photoabsorption band because it receives photons in the near infrared region, where the sunlight energy density is the highest. In addition, it is carefully adjusted to the bottom of the CB to minimize the voltage loss in the experiments. The first bands of the cationic dyes of T4 or AZOL are dipole-forbidden, so we instead examined the next bands around 2 eV. As shown in Table 6, dye adsorption increased the energy of the peak by 0.11-0.18 eV. The change is the smallest in S(m), the largest in A(mm), and in between in C(bm) or C(mm). These changes show a positive correlation with those of the HOMO-LUMO gap calculated by the dyed slab model, as shown in Tables 4 and 5. Dye adsorption increased the gaps with respect to the desorbed dyes by about 0.04 eV in S(m), 0.10 eV in A(mm), 0.05 eV in C(mm), and 0.08 eV in C(bm), see the last column of Table 6. Adsorption also changed the oscillator strength of the band. Although the strengths for each mode fluctuate, we observed about 15 % increases in S(m) and C(bm), a 15 % decrease in A(mm), and 30 % increase in C(mm). 23



(a): BD/(TiO2 )24

(b): BD/(TiO2 )48

(d): BD/(TiO2 )48 (top view)

(c): BD/(TiO2 )24 (top view)

Figure 6: Structure of the cluster model of the dyed anatase surface. The oxygen atoms on the cluster edge are shown in gray.

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Table 6: Position of the First Photoabsorption Band in eV (0.05 eV Gaussian width). Structure Desorbed S(m) A(mm) C(mm) C(bm) Average

BD PRT12 PPY ACAC TF2 T4a AZOLa 1.45 1.48 1.43 1.52 1.66 1.86 1.87 b b b 0.10 0.12 0.11 0.14 0.14 0.19 0.19 0.17 0.22 0.14 0.15 0.12 0.22 0.16 0.13 0.16 0.12 0.14 0.13 0.18 0.16 0.15 0.17 0.13 0.13 0.12 0.19 0.15 0.14 0.17 0.13 0.14 a second photoabsorption band. b not calculated.

Average 0.12 0.17 0.15 0.15

Table 7: Sum of the Oscillator Strength multiplied by Mulliken Populations on the TiO2 Cluster of Excited States below Ethres (×10−2 atomic units). Structure BD Ethres 2.3 S(m) 8.49 A(mm) 4.02 C(mm) 10.47 C(bm) 11.10 Average 8.52

PRT12 2.4 a

8.28 12.40 11.15 10.61

PPY ACAC TF2 2.4 2.5 2.5 7.88 5.47 2.49 7.34 3.42 1.64 13.42 10.94 5.48 11.27 9.34 3.41 9.98 7.29 3.25 a not calculated.

T4 2.5

AZOL Average 2.6 a a 6.08 7.23 3.05 5.00 13.33 11.37 11.06 10.16 10.37 9.54 10.24 8.26

Rapid injection is expected if a virtual orbital that accommodates the excited electron has a large population on the anchoring carboxy group; for example, tpy-based cyclometalated Ru complexes, that only differ in the position of the Ru–C bond, were reported to have very different efficiencies, 87 which was attributed to the different LUMO populations on the anchoring group. Hence, we compared the number of excited electrons in the TiO2 slab, multiplied by the photoabsorption strength of the transition. More specifically, we calculated the sum of the Mulliken populations pi on the slab atoms multiplied by the oscillator strength fi of the ith excited state. As many excited states appeared the cumulative value below the energy Ethres was convenient for analysis. ∑

P (Ethres ) =

p i fi

(13)

ϵi C(mm) ≈ C(bm) > S(m). A(mm) has the largest excess energy, but because of the twist in the anchoring carboxy group, the actual injection is very slow. On the other hand,

31



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Table 8: Energy Differences between the First Photoabsorption Band and the Bottom of the CB (eV). Structure BD PRT12 PPY ACAC TF2 T4a AZOLa b b b S(m) -0.12 -0.12 -0.23 -0.22 A(mm) 0.12 0.32 0.15 0.00 0.03 0.16 0.29 C(mm) 0.00 0.22 -0.01 -0.12 -0.10 0.07 0.19 C(bm) 0.03 0.13 -0.04 -0.16 -0.14 0.04 0.13 Average 0.01 0.22 -0.01 -0.13 -0.11 0.09 0.20 a b second photoabsorption band. not calculated.

Average -0.17 0.15 0.04 0.00

the first photoabsorption band of the singly-anchored S(m) structure was below the CB. This fact implies the importance of the two anchoring groups on the terpyridine ligand. Suppose that we replace the carboxy groups at the 4 and 4′′ positions on the ligand with an electronegative, non-anchoring group, such as a fluorine atom. This family of dyes would adsorb on the surface with the S(m) mode, have a small excess energy, and show a slow injection. Among the seven dyes studied, PRT12 had the largest excess energy. The second largest value, which was obtained for AZOL, actually corresponds to the second band because the first band has a very small oscillator strength. The first bands of BD or PPY in the most stable C(mm) structures are very close to the bottom of the CB, and those of ACAC or TF2 are somewhat (about 0.1 eV) below it. Without the excess energy, injection would take place through intersystem crossing to the triplet excited state, followed by slow electron tunneling to the trap level in the band gap. 89 The message from eq. 7 is that the HOMO energy of the dye is a very important parameter for the injection rate, which has clear lower and upper limits. It was experimentally demonstrated that an upper limit of the dye’s redox potential existed for the regeneration reaction between dye cation and triiodide anion to take place. 89 On the other hand, the deeper HOMO energy reduces the overlap between photoabsorption and CB DOS, resulting in an injection loss. The LUMO energy of the desorbed dye is often compared with the bottom of the CB of anatase to predict the injection ability. However, this only gives an approximate

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value because of relaxation effects and the large configuration mixing in organo-ruthenium dyes. Note also that the LUMO is sensitive to computational details, such as the basis set or the functional used. Table 9: Sum of the Imaginary part of the Energy multiplied by the Oscillator Strength (×10−3 atomic units). Structure Ethres S(m) A(mm) C(mm) C(bm) Average

BD PRT12 PPY ACAC TF2 T4 AZOL Average 2.3 2.4 2.4 2.5 2.5 2.5 2.6 a a a 0.90 1.07 0.69 0.23 0.72 0.83 2.80 1.02 0.85 0.27 1.98 0.97 1.24 2.89 6.73 3.33 2.41 1.50 3.28 2.72 3.27 2.50 2.72 2.35 1.54 0.07 1.47 1.48 1.73 1.78 4.08 1.94 1.37 0.52 2.24 1.72 a not calculated.

As a quantitative measure of the injection ability of the excited state we calculated the oscillator strength multiplied by the imaginary part of the energy. Under the given light intensity, the oscillator strength is proportional to the number of excited molecules, while the imaginary part is proportional to the average number of transferred electrons from an excited state per time unit. We compared the sums of these parameters below Ethres , and as seen in Table 9, the value obtained for the most stable C(mm) structure was the largest among the adsorption modes of all the studied dyes. On the other hand, small values were observed for S(m), due to the lack of excess energy, and for A(mm), due to the twist in the anchoring group. The TF2 dye showed the smallest value among all because its strong photoabsorption peak was calculated slightly below the bottom of the CB, as seen in Table 8, due to the deepest HOMO (Table 4). Considering that the state lifetime is sensitive to the position of the CB, optimization of the electrolyte composition may drastically improve its value. PRT12 showed the largest value in Table 9. This dye exhibits broad photoabsorption in the low energy region as well as relatively shallow HOMO (Table 4), which contributes to the improved injection ability. Note that the experimentally determined injection efficiency of PRT12 was 85 %, which was larger than that of BD (52 %) under the same conditions. 77 It is interesting to compare the results of Tables 7 and 9. The simpler method applied 33



in Table 7 agreed qualitatively with the more elaborated method of Table 9: the ordering of the dyes was, in average, almost reproduced. However, there are visible differences: In Table 9, PRT12-C(mm) exhibits the largest value, twice as large as those of PPY-C(mm) and T4-C(mm), but in Table 7, the values are almost the same. Finally, we list the limitations of the current study: The proposed method to calculate the molecular resonance state on a surface is quite general and is applicable to other systems such as molecular or spintronic devices, although in this paper we applied it to DSSCs. For simplicity, only the perturbation correction to the imaginary part of the energy was calculated using eq. 7. The real part of the energy correction may not be small, and higherorder or self-consistent calculations may be required. Because of the computational cost, a cluster model was used to calculate the unperturbed excited states. We focused on electron injection from the singlet excited state to the CB. Another important process, namely, the injection from the triplet state 89 was not covered. We assumed that the dye forms a regular monolayer with experimentally observed coverage on a perfect anatase (101) surface. Other structures, such as dimer formation, would be important for understanding the energy loss mechanism, in particular for organic dyes.

9

Conclusions

The ab initio method was applied to examine metastable molecular excited states on a solid surface. These excited states play an important role in molecular or spintronic devices, DSSCs, and hydrogen-generating devices. Using the resonance state theory, a formula was presented to calculate the decay rate of the excited state through electron injection to the CB, and the electron injection process in DSSCs was analyzed. We reported stable adsorption structures for a series of Ru dyes on an anatase (101) surface. The relative stabilities of various adsorption structures of BD analogs were similar, implying that other metal dctpy complexes would bind to the surface in the same way. The first photoabsorption band of the

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dyes was found to be close to the bottom of the CB, and as a result, the position of the latter governed the injection rate. We carefully calculated the CB DOS using a two-dimensional slab model, the B3LYP functional, and PB solvation model. Without the implicit solvation model, the surface dipole and the position of the DOS were very sensitive to the structure and therefore not reliable. The effects of molecular structure and adsorption mode on the HOMO and LUMO energies, excitation energies, photoabsorption strength, population of excited electrons, and electron injection rate were reported. Overall, PRT12 (in its most stable structure) showed the best predicted efficiency, which is consistent with previous experiments.

Acknowledgement We acknowledge Prof. Satoru Iuchi at Nagoya University for his valuable discussions. The computations were partially conducted using facilities of the Institute for Molecular Science, Okazaki, Japan. This work has been supported in part by a Grant-in-Aid for Scientific Research from the Computational Materials Science Initiative (CMSI), Japan, and by the joint usage / research program of the Institute of Materials and Systems for Sustainability (IMaSS), Nagoya University.

Supporting Information Available The coordinates of the low-energy structures in Figure 4 and Table 2 are provided. This information is available free of charge via the Internet at http://pubs.acs.org

References (1) Wu, S.; Ogawa, N.; Ho, W. Atomic-scale coupling of photons to single-molecule junctions. Science 2006, 312, 1362–1365. 35



(2) Steil, S.; Grosmann, N.; Laux, M.; Ruffing, A.; Steil, D.; Wiesenmayer, M.; Mathias, S.; Monti, O.; Cinchetti, M.; Aeschlimann, M. Spin-dependent trapping of electrons at spinterfaces. Nature Physics 2013, 9, 242–247. (3) Grätzel, M. Dye-sensitized solar cells. J. Photoch. Photobio. C 2003, 4, 145–153. (4) Youngblood, W.; Lee, S.; Maeda, K.; Mallouk, T. Visible Light Water Splitting Using Dye-Sensitized Oxide Semiconductors. Acc. Chem. Res. 2009, 42, 1966–1973. (5) Murdoch, M.; Waterhouse, G.; Nadeem, M.; Metson, J.; Keane, M.; Howe, R.; Llorca, J.; Idriss, H. The effect of gold loading and particle size on photocatalytic hydrogen production from ethanol over Au/TiO2 nanoparticles. Nature Chemistry 2011, 3, 489–492. (6) Silva, C.; Juarez, R.; Marino, T.; Molinari, R.; Garcia, H. Influence of Excitation Wavelength (UV or Visible Light) on the Photocatalytic Activity of Titania Containing Gold Nanoparticles for the Generation of Hydrogen or Oxygen from Water. J. Am. Chem. Soc. 2011, 133, 595–602. (7) Lee, M.; Teuscher, J.; Miyasaka, T.; Murakami, T.; Snaith, H. Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites. Science 2012, 338, 643–647. (8) Hu, K.; Blair, A.; Piechota, E.; Schauer, P.; Sampaio, R.; Parlane, F.; Meyer, G.; Berlinguette, C. Kinetic pathway for interfacial electron transfer from a semiconductor to a molecule. Nature Chemistry 2016, 8, 853–859. (9) Zigler, D.; Morseth, Z.; Wang, L.; Ashford, D.; Brennaman, M.; Grumstrup, E.; Brigham, E.; Gish, M.; Dillon, R.; Alibabaei, L.; Meyer, G.; Meyer, T.; Papanikolas, J. Disentangling the Physical Processes Responsible for the Kinetic Complexity in Interfacial Electron Transfer of Excited Ru(II) Polypyridyl Dyes on TiO2 . J. Am. Chem. Soc. 2016, 138, 4426–4438. 36


Page 36 of 55

Page 37 of 55 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


(10) Fano, U. Effects of Configuration Interaction on Intensities and Phase Shifts. Phys. Rev. 1961, 124, 1866–1878. (11) Stier, W.; Prezhdo, O. V. Nonadiabatic molecular dynamics simulation of light-induced electron transfer from an anchored molecular electron donor to a semiconductor acceptor. J. Phys. Chem. B 2002, 106, 8047–8054. (12) Duncan, W. R.; Prezhdo, O. V. Nonadiabatic molecular dynamics study of electron transfer from alizarin to the hydrated Ti4+ ion. J. Phys. Chem. B 2005, 109, 17998– 18002. (13) Duncan, W. R.; Stier, W. M.; Prezhdo, O. V. Ab initio Nonadiabatic Molecular Dynamics of the Ultrafast Electron Injection across the Alizarin-TiO2 Interface. J. Am. Chem. Soc. 2005, 127, 7941–7951. (14) Duncan, W. R.; Prezhdo, O. V. Theoretical studies of photoinduced electron transfer in dye-sensitized TiO2 . Annu. Rev. Phys. Chem. 2007, 58, 143–184. (15) Wei Ma, W.; Jiao, Y.; Meng, S. Predicting Energy Conversion Efficiency of Dye Solar Cells from First Principles. J. Phys. Chem. C 2014, 118, 16447–16457. (16) Li, J.; Nilsing, M.; Kondov, I.; Wang, H.; Persson, P.; Lunell, S.; Thoss, M. Dynamical simulation of Photoinduced electron transfer Reactions in Dye- Semiconductor systems with different Anchor groups. J. Phys. Chem. C 2008, 112, 12326–12333. (17) Jones, D. R.; Troisi, A. A method to rapidly predict the charge injection rate in dye sensitized solar cells. Phys. Chem. Chem. Phys. 2010, 12, 4625–4634. (18) Martsinovich, N.; Troisi, A. Theoretical studies of dye-sensitised solar cells: From electronic structure to elementary processes. Energy Environ. Sci. 2011, 4, 4473–4495. (19) Martsinovich, N.; Troisi, A. High-Throughput Computational Screening of Chromophores for Dye-Sensitized Solar Cells. J. Phys. Chem. C 2011, 115, 11781–11792. 37



(20) Martsinovich, N.; Ambrosio, F.; Troisi, A. Adsorption and electron injection of the N3 metal–organic dye on the TiO2 rutile (110) surface. Phys. Chem. Chem. Phys. 2012, 14, 16668–16676. (21) Siegert, A. J. On the derivation of the dispersion formula for nuclear reactions. Phys. Rev. 1939, 56, 750–752. (22) Sulzer, D.; Iuchi, S.; Yasuda, K. A New Method To Evaluate Excited States Lifetimes Based on Green’s Function: Application to Dye-Sensitized Solar Cells. J. Chem. Theory Comput. 2016, 12, 3074–3086. (23) Landau, L. D.; Lifshitz, E. M. Quantum Mechanics, third edition, Chapter 17, section 134 ; Pergamon Press, 1965. (24) Zeldovich, Y. B. On the theory of unstable states. Soviet Physics JETP 1961, 12, 542–545. (25) Moiseyev, N. Quantum theory of resonances: calculating energies, widths and crosssections by complex scaling. Physics Reports 1998, 302, 212–293. (26) Riss, U. V.; Meyer, H. D. Calculation of resonance energies and widths using the complex absorbing potential method. J. Phys. 1993, B26, 4503–4536. (27) Ehara, M.; Sommerfeld, T. CAP/SAC-CI method for calculating resonance states of metastable anions. Chemical Physics Letters 2012, 537, 107–112. (28) Runge, E.; Gross, E. K. U. Density-Functional Theory for Time-Dependent Systems. Phys. Rev. Lett. 1984, 52, 997–1000. (29) Fetter, A. L.; Walecka, J. D. Quantum theory of many-particle systems, Chapter 15, section 60 ; International series in pure and applied physics; McGraw-Hill, 1971. (30) Whitenack, D.; Wasserman, A. Density functional resonance theory of unbound electronic systems. Phys Rev Lett. 2011, 107, 163002. 38


Page 38 of 55

Page 39 of 55 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


(31) Onida, G.; Reining, L.; Rubio, A. Electronic excitations: density-functional versus many-body Green’s-function approaches. Rev. Mod. Phys. 2002, 74, 601–659. (32) Landau, L. D.; Lifshitz, E. M. Quantum Mechanics, third edition, Chapter 6, section 43 ; Pergamon Press, 1965. (33) Perdew, J. P.; Levy, M. Physical content of the exact Kohn-Sham orbital energies: band gaps and derivative discontinuities. Phys. Rev. Lett. 1983, 51, 1884. (34) Akimov, A. V.; Neukirch, A. J.; Prezhdo, O. V. Theoretical Insights into Photoinduced Charge Transfer and Catalysis at Oxide Interfaces. Chem. Rev. 2013, 113, 4496–4565. (35) Rowe, D. Equations-of-Motion Method and the Extended Shell Model. Rev. Mod. Phys. 1968, 40, 153–166. (36) Péchy, P.; Renouard, T.; Zakeeruddin, S. M.; Humphry-Baker, R.; Comte, P.; Liska, P.; Cevey, L.; Costa, E.; Shklover, V.; Spiccia, L.; Deacon, G. B.; Bignozzi, C. A.; Grätzel, M. Engineering of Efficient Panchromatic Sensitizers for Nanocrystalline TiO2 Based Solar Cells. J. Am. Chem. Soc. 2001, 123, 1613–1624. (37) Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 1993, 47, R558–R561. (38) Kresse, G.; Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal– amorphous-semiconductor transition in germanium. Phys. Rev. B 1994, 49, 14251– 14269. (39) Hafner, J. Ab-initio simulations of materials using VASP: Density-functional theory and beyond. J. Comput. Chem. 2008, 29, 2044–2078. (40) Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 1994, 50, 17953–17979. (41) Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 1999, 59, 1758–1775. 39



(42) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865–3868. (43) Becke, A. D. Density functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648–5652. (44) Pastore, M.; Fantacci, S.; De Angelis, F. Modeling Excited States and Alignment of Energy Levels in Dye-Sensitized Solar Cells: Successes, Failures, and Challenges. J. Phys. Chem. C 2013, 117, 3685–3700. (45) Mathew, K.; Sundararaman, R.; Letchworth-Weaver, K.; Arias, T. A.; Hennig, R. G. Implicit solvation model for density-functional study of nanocrystal surfaces and reaction pathways. J. Chem. Phys. 2014, 140, 084106. (46) Tomasi, J.; Mennucci, B.; Cammi, R. Quantum mechanical continuum solvation models. Chem. Rev. 2005, 105, 2999–3094. (47) 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.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannen¨ Foresman, J. B.; Ortiz, J. V.; berg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, O.;

40


Page 40 of 55

Page 41 of 55 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


Cioslowski, J.; Fox, D. J. Gaussian 09 Revision C.01. 2009; Gaussian Inc. Wallingford CT 2009. (48) Kislenko, S. A.; Amirov, R. H.; Samoylov, I. S. Effect of Cations on the TiO2 /Acetonitrile Interface Structure: A Molecular Dynamics Study. J. Phys. Chem. C 2013, 117, 10589–10596. (49) Tateyama, Y.; Sumita, M.; Ootani, Y.; Aikawa, K.; Jono, R.; Han, L.; Sodeyama, K. Acetonitrile Solution Effect on Ru N749 Dye Adsorption and Excitation at TiO2 Anatase Interface. J. Phys. Chem. C 2014, 118, 16863–16871. (50) Spreafico, C.; Schiffmann, F.; VandeVondele, J. Structure and Mobility of Acetic Acid at the Anatase (101)/Acetonitrile Interface. J. Phys. Chem. C 2014, 118, 6251–6260. (51) Hehre, W. J.; Ditchfield, R.; Stewart, R. F.; Pople, J. A. Self-Consistent Molecular Orbital Methods. IV. Use of Gaussian Expansions of Slater-Type Orbitals. Extension to Second-Row Molecules. J. Chem. Phys. 1970, 52, 2769–2773. (52) Hehre, W. J.; Ditchfield, R.; Pople, J. A. Self-consistent molecular-orbital methods. 12. Further extensions of gaussian-type basis sets for use in molecular-orbital studies of organic molecules. J. Chem. Phys. 1972, 56, 2257–2261. (53) Rassolov, V. A.; Pople, J. A.; Ratner, M. A.; Windus, T. L. 6-31G* basis set for atoms K through Zn. J. Chem. Phys. 1998, 109, 1223–1229. (54) Nakai, H.; Heyd, J.; Scuseria, G. E. Periodic-Boundary-Condition Calculation using Heyd-Scuseria-Ernzerhof Screened Coulomb Hybrid Functional: Electronic Structure of Anatase and Rutile TiO2 . J. Comput. Chem. Jpn. 2006, 5, 7–18. (55) Wadt, W. R.; Hay, P. J. Ab initio effective core potentials for molecular calculations. Potentials for main group elements Na to Bi. J. Chem. Phys. 1985, 82, 284–298.

41



(56) Hay, P. J.; Wadt, W. R. Ab initio effective core potentials for molecular calculations. Potentials for the transition metal atoms Sc to Hg. J. Chem. Phys. 1985, 82, 270–283. (57) Hay, P. J.; Wadt, W. R. Ab initio effective core potentials for molecular calculations. Potentials for K to Au including the outermost core orbitals. J. Chem. Phys. 1985, 82, 299–310. (58) Umerski, A. Closed-form solutions to surface Green’s functions. Phys. Rev. B 1997, 55, 5266–5275. (59) Reuter, M. G.; Boffi, N. M.; Ratner, M. A.; Seideman, T. The role of dimensionality in the decay of surface effects. J. Chem. Phys. 2013, 138, 084707. (60) Reuter, M. G.; Seideman, T.; Ratner, M. A. Probing the surface-to-bulk transition: A closed-form constant-scaling algorithm for computing subsurface Green functions. Phys. Rev. B 2011, 83, 085412. (61) Shklover, V.; Ovchinnikov, Y. E.; Braginsky, L. S.; Zakeeruddine, S. M.; Gräzel, M. Structure of Organic/Inorganic Interface in Assembled Materials Comprising Molecular Components. Crystal Structure of the Sensitizer Bis [(4,4 -carboxy-2,2 -bipyridine) (thiocyanato)] ruthenium(II). Chem. Mater. 1998, 10, 2533–2541. (62) Bauer, C.; Boschloo, G.; Mukhtar, E.; Hagfeldt, A. Interfacial Electron-Transfer Dynamics in Ru(tcterpy)(NCS)3 -Sensitized TiO2 Nanocrystalline Solar Cells. J. Phys. Chem. B 2002, 106, 12693–12704. (63) Nazeeruddin, M. K.; Humphry-Baker, R.; Liska, P.; Grätzel, M. Investigation of Sensitizer Adsorption and the Influence of Protons on Current and Voltage of a DyeSensitized Nanocrystalline TiO2 Solar Cell. J. Phys. Chem. B 2003, 107, 8981–8987. (64) Finnie, K. S.; Bartlett, J. R.; Woolfrey, L. Vibrational Spectroscopic Study of the

42


Page 42 of 55

Page 43 of 55 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


Coordination of (2,2′ -Bipyridyl- 4,4′ -dicarboxylic acid) ruthenium(II) Complexes to the Surface of Nanocrystalline Titania. Langmuir 1998, 14, 2744–2749. (65) León, C. P.; Kador, L.; Peng, B.; Thelakkat, M. Characterization of the Adsorption of Ru-bpy Dyes on Mesoporous TiO2 Films with UV-Vis, Raman, and FTIR Spectroscopies. J. Phys. Chem. B 2006, 110, 8723–8730. (66) Persson, P.; Lundqvist, M. J. Calculated Structural and Electronic Interactions of the Ruthenium Dye N3 with a Titanium Dioxide Nanocrystal. J. Phys. Chem. B 2005, 109, 11918–11924. (67) De Angelis, F.; Fantacci, S.; Selloni, A.; Nazeeruddin, M. K.; Grätzel, M. Timedependent density functional theory investigations on the excited states of Ru (II)dye-sensitized TiO2 nanoparticles: the role of sensitizer protonation. J. Am. Chem. Soc. 2007, 129, 14156–14157. (68) De Angelis, F.; Fantacci, S.; Selloni, A.; Grätzel, M.; Nazeeruddin, M. K. Influence of the Sensitizer Adsorption Mode on the Open-Circuit Potential of Dye-Sensitized Solar Cells. Nano Lett. 2007, 7, 3189–3195. (69) De Angelis, F.; Fantacci, S.; Selloni, A. Alignment of the Dye’s Molecular Levels with the TiO2 Band Edges in Dye-Sensitized Solar Cells: a DFT-TDDFT Study. Nanotechnology 2008, 19, 424002. (70) Schiffmann, F.; VandeVondele, J.; Hutter, J.; Wirz, R.; Urakawa, A.; Baiker, A. Protonation-Dependent Binding of Ruthenium Bipyridyl Complexes to the Anatase (101) Surface. J. Phys. Chem. C 2010, 114, 8398–8404. (71) Sodeyama, K.; Sumita, M.; O’Rourke, C.; Terranova, U.; Islam, A.; Han, L.; Bowler, D. R.; Tateyama, Y. Protonated Carboxyl Anchor for Stable Adsorption of Ru N749 Dye (Black Dye) on a TiO2 Anatase (101) Surface. J. Phys. Chem. Lett. 2012, 3, 472–477. 43



(72) Fantacci, S.; Lobello, M. G.; Angelis, F. D. Everything you always wanted to know about black dye (but were afraid to ask): A DFT/TDDFT investigation. Chimia (Aarau) 2013, 67, 121–128. (73) Srimath, K.; Ajay, R.; Fantacci, S.; Guarnera, S.; Polli, D.; Lanzani, G.; De Angelis, F.; Petrozza, A. Role of hot singlet excited states in charge generation at the black dye/TiO2 interface. ACS Appl. Mater. Interfaces 2013, 5, 4334–4339. (74) Salvatori, P.; Agrawal, S.; Barreddi, C.; Malapaka, C.; de Borniold, M.; De Angelis, F. Stability of ruthenium/organic dye co-sensitized solar cells: a joint experimental and computational investigation. RSC Advances 2014, 101, 57620–57628. (75) Srinivas, K.; Yesudas, K.; Bhanuprakash, K.; Rao, V. J.; Giribabu, L. A Combined Experimental and Computational Investigation of Anthracene Based Sensitizers for DSSC: Comparison of Cyanoacrylic and Malonic Acid Electron Withdrawing Groups Binding onto the TiO2 Anatase (101) Surface. J. Phys. Chem. C 2009, 113, 20117– 20126. (76) Yang, S.-H.; Wu, K.-L.; Chi, Y.; Cheng, Y.-M.; Chou, P.-T. Tris(thiocyanate) Ruthenium(II) Sensitizers with Functionalized Dicarboxyterpyridine for Dye-Sensitized Solar Cells. Angewandte Chemie, International Edition 2011, 50, 8270–8274. (77) Lin, H.-W.; Wang, Y.-S.; Huang, Z.-Y.; Lin, Y.-M.; Chen, C.-W.; Yang, S.-H.; Wu, K.L.; Chi, Y.; Liu, S.-H.; Chou, P.-T. Origins of device performance in dicarboxyterpyridine Ru(II) dye-sensitized solar cells. Physical Chemistry Chemical Physics 2012, 14, 14190–14195. (78) Funaki, T.; Yanagida, M.; Onozawa-Komatsuzaki, N.; Kasuga, K.; Kawanishi, Y.; Kurashige, M.; Sayama, K.; Sugihara, H. Synthesis of a new class of cyclometallated ruthenium(II) complexes and their application in dye-sensitized solar cells. Inorganic Chemistry Communications 2009, 12, 842–845. 44


Page 44 of 55

Page 45 of 55 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


(79) Funaki, T.; Otsuka, H.; Onozawa-Komatsuzaki, N.; Kasuga, K.; Sayama, K.; Sugihara, H. Systematic evaluation of HOMO energy levels for efficient dye regeneration in dye-sensitized solar cells. Journal of Materials Chemistry A: Materials for Energy and Sustainability 2014, 2, 15945–15951. (80) Islam, A.; Chowdhury, F. A.; Chiba, Y.; Komiya, R.; Fuke, N.; Ikeda, N.; Nozaki, K.; Han, L. Synthesis and Characterization of New Efficient Tricarboxyterpyridyl (βdiketonato) Ruthenium(II) Sensitizers and Their Applications in Dye-Sensitized Solar Cells. Chem. Mater. 2006, 18, 5178–5185. (81) Chou, C.-C.; Wu, K.-L.; Chi, Y.; Hu, W.-P.; Yu, S. J.; Lee, G.-H.; Lin, C.-L.; Chou, P.T. Ruthenium(II) Sensitizers with Heteroleptic Tridentate Chelates for Dye-Sensitized Solar Cells. Angew. Chem. Int. Ed. 2011, 50, 2054–2058. (82) Yoneda, E.; Grätzel, M.; Nazeeruddin, M. K. Eur. Pat. Appl. 2009, EP 2036955. (83) Schulze, B.; Brown, D. G.; Robson, K. C. D.; Friebe, C.; Jäger, M.; Birckner, E.; Berlinguette, C. P.; Schubert, U. S. Cyclometalated Ruthenium(II) Complexes Featuring Tridentate Click-Derived Ligands for Dye-Sensitized Solar Cell Applications. Chem. Eur. J. 2013, 19, 14171–14180. (84) Jackson, J. Classical Electrodynamics, Chapter 1, section 1.6 ; Hamilton Printing Company, 1998. (85) Martin, R. M. Electronic Structure, Basic Theory and Practical Methods, Appendix F ; Cambridge University Press, 2004. (86) Norskov, J.; Rossmeisl, J.; Logadottir, A.; Lindqvist, L. Origin of the Overpotential for Oxygen Reduction at a Fuel-Cell Cathode. J. Phys. Chem. B 2004, 108, 17886–17892. (87) Wadman, S. H.; Kroon, J. M.; Bakker, K.; Havenith, R. W. A.; van Klink, G. P. M.;

45



van Koten, G. Cyclometalated Organoruthenium Complexes for Application in DyeSensitized Solar Cells. Organometallics 2010, 29, 1569–1579. (88) Dreuw, A.; Head-Gordon, M. Single-Reference ab Initio Methods for the Calculation of Excited States of Large Molecules. Chem. Rev. 2005, 105, 4009–4037. (89) Listorti, A.; O’Regan, B.; Durrant, J. R. Electron transfer dynamics in dye-sensitized solar cells. Chem. Mater. 2011, 23, 3381–3399.

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Figure 1: Comparison of the PCM solvation energies in acetonitrile with different implementations (kJ/mol).

Figure 2: Structure of the black dye analogs.



Page 48 of 55

Ti5c

C

5.35

X 7.57

O3c 7.57 O2c B A Figure 3: Anchoring points on the anatase (101) surface.

Table 1: Relative Energies (kJ/mol) and Selected Bond Lengths (˚ A) of BD Adsorbed on an Anatase (101) Surface. Length S(m) S(b) A(mm) B(mm) C(mm) C(bm) C(bm)H+ Desorbed sub 20.63 31.64 0.00 18.17 36.62 108.38 ∆E 27.72 82.42 r(O–O) 6.47 6.44 7.14 7.24 5.89 6.12 6.10 6.38 6.69 7.00 6.48 6.43 6.90 6.97 6.87 6.75 r(O–Ti) 2.15 2.09 2.17 1.93 2.00 2.00 2.08 2.28 2.17 2.01 2.03 2.13 2.09 2.13 2.00 r(O–H) 1.48 1.62a 1.37 1.78 1.35 1.49 1.39 1.55 1.68a 1.42 1.88 1.63 a Hydrogen bond length toward neighbor dye.

Dye PRT12 PPY ACAC TF2 T4 AZOL

Table 2: Relative Energies (kJ/mol) of S(m) A(mm) B(mm) C(mm) 31.65 25.33 27.16 0.00 28.46 23.13 24.92 0.00 32.80 25.45 33.40 0.00 30.69 24.99 33.89 0.00 14.37 32.80 15.82 0.00 19.90 23.43 18.10 0.00

the BD Analogs Adsorbed on an Anatase (101) Surface. C(bm) C(bm)H+ sub 23.66 38.00 19.12 31.16 20.29 36.69 20.18 35.80 2.38 16.71 7.49 21.63

Table 3: HOMO Energies of Structural Isomers (eV). All the Methods Contain Nine CH3 CN Molecules as the Solvent. Without Implicit Solvation, Small Structural Changes Can Induce Large Fluctuations. Molecule BD+N(CH3 )+ PPY 4 Functional PBE PBE B3LYP B3LYP PBE B3LYP Solvent vacuum PCM PB vacuum PB PB -1.87 -1.42 -2.50 -3.25 -1.42 -2.50 Isomer A Isomer B -1.28 -1.46 -2.50 -2.54 -1.39 -2.60


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6.47 2.15

1.48

1.68

1.68

1.63

1.63 2.28

1.55

2.09

b: S(b), 82.4 kJ a: S(m), 27.7 kJ

2.17 1.37

2.17 7.14 1.42

1.88

7.24

2.01

1.78

1.93

c: A(mm), 20.6 kJ d: B(mm), 31.6 kJ Figure 4: Adsorption structure of BD on anatase (101) surface, relative energy (kJ/mol), and selected atomic distances (˚ A). Black: O–O′ , green: Ti–O, blue: hydrogen bonded O–H.

Table 4: HOMO Energy of Structure BD PRT12 PPY Desorbed -5.31 -5.23 -5.28 a S(m) -0.85 -0.90 A(mm) -0.66 -0.66 -0.69 C(mm) -0.75 -0.75 -0.80 C(bm) -0.74 -0.81 -0.85 Average -0.75 -0.73 -0.81 a not calculated.

a Dye in ACAC -5.54 -0.82 -0.64 -0.72 -0.78 -0.74

Various Adsorption Modes (Shift from Desorbed in eV). TF2 T4 AZOL Average Average (cluster) -5.63 -5.59 -5.55 a a -0.88 -0.86 0.25 -0.71 -0.74 -0.68 -0.68 0.32 -0.78 -0.81 -0.76 -0.76 0.26 -0.83 -0.86 -0.82 -0.81 0.27 -0.80 -0.80 -0.75 -0.77 0.28



2.00 1.63

5.89

2.03 1.35

2.13

6.10

6.12

f: C(bm), 18.2 kJ

e: C(mm), 0.0 kcal

2.09 2.08

Page 50 of 55

2.00 1.39

g: C(bm)H+ sub , 36.6 kJ

Structure BD Desorbed -3.14 S(m) -0.83 A(mm) -0.60 C(mm) -0.73 C(bm) -0.68 Average -0.71 a not calculated.

PRT12 -2.99 a

-0.53 -0.72 -0.71 -0.65

Table 5: LUMO Energy of the PPY ACAC TF2 T4 -3.00 -3.22 -3.12 -3.14 a -0.83 -0.79 -0.82 -0.58 -0.57 -0.62 -0.62 -0.71 -0.68 -0.71 -0.72 -0.76 -0.71 -0.75 -0.75 -0.72 -0.68 -0.73 -0.70

Adsorbed Dyes (eV). AZOL Average -3.11 a -0.82 -0.59 -0.58 -0.69 -0.71 -0.73 -0.73 -0.67 -0.69


2.00 1.49

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Figure 5: Comparison of the total DOS of dyed slab models. The inset shows a magnification near the bottom of the CB.

Table 6: Position of the First Photoabsorption Band in eV (0.05 eV Gaussian width). Structure BD PRT12 PPY ACAC TF2 T4a AZOLa Average 1.48 1.43 1.52 1.66 1.86 1.87 Desorbed 1.45 b b b S(m) 0.10 0.12 0.11 0.14 0.12 A(mm) 0.14 0.19 0.19 0.17 0.22 0.14 0.15 0.17 C(mm) 0.12 0.22 0.16 0.13 0.16 0.12 0.14 0.15 C(bm) 0.13 0.18 0.16 0.15 0.17 0.13 0.13 0.15 Average 0.12 0.19 0.15 0.14 0.17 0.13 0.14 a second photoabsorption band. b not calculated.

Table 7: Sum of the Oscillator Strength multiplied by below Ethres (×10−2 atomic units). Structure BD PRT12 PPY ACAC TF2 2.4 2.4 2.5 2.5 Ethres 2.3 a S(m) 8.49 7.88 5.47 2.49 A(mm) 4.02 8.28 7.34 3.42 1.64 C(mm) 10.47 12.40 13.42 10.94 5.48 C(bm) 11.10 11.15 11.27 9.34 3.41 Average 8.52 10.61 9.98 7.29 3.25 a not calculated.

Mulliken Populations on the TiO2 Cluster of Excited States T4 2.5

AZOL 2.6

Average

a

a

7.23 13.33 10.16 10.24

3.05 11.37 10.37 8.26

6.08 5.00 11.06 9.54



(a): BD/(TiO2 )24

(c): BD/(TiO2 )24 (top view)

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(b): BD/(TiO2 )48

(d): BD/(TiO2 )48 (top view)

Figure 6: Structure of the cluster model of the dyed anatase surface. The oxygen atoms on the cluster edge are shown in gray.

Table 8: Energy Differences between the First Photoabsorption Band and the Bottom of the CB (eV). Structure BD PRT12 PPY ACAC TF2 T4a AZOLa Average b b b -0.23 -0.22 -0.17 S(m) -0.12 -0.12 A(mm) 0.12 0.32 0.15 0.00 0.03 0.16 0.29 0.15 C(mm) 0.00 0.22 -0.01 -0.12 -0.10 0.07 0.19 0.04 C(bm) 0.03 0.13 -0.04 -0.16 -0.14 0.04 0.13 0.00 Average 0.01 0.22 -0.01 -0.13 -0.11 0.09 0.20 a second photoabsorption band. b not calculated.


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Figure 7: Left shows MO φbot (r), whose orbital energy is at the bottom of the CB. Right shows f (r) in eq. 15, that displays the spatial distribution of SE.

Figure 8: Imaginary part of the self-energy (in atomic units) and DOS (arbitrary units) of the anatase (101) surface near the bottom of the CB.



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Figure 9: Oscillator strength, excited-state lifetime, and DOS for the cluster and slab models.

Table 9: Sum of the Imaginary part of the Energy multiplied by the Oscillator Strength (×10−3 atomic units). Structure BD PRT12 PPY ACAC TF2 T4 AZOL Average 2.4 2.4 2.5 2.5 2.5 2.6 Ethres 2.3 a a a S(m) 0.90 1.07 0.69 0.23 0.72 A(mm) 0.83 2.80 1.02 0.85 0.27 1.98 0.97 1.24 C(mm) 2.89 6.73 3.33 2.41 1.50 3.28 2.72 3.27 C(bm) 2.50 2.72 2.35 1.54 0.07 1.47 1.48 1.73 Average 1.78 4.08 1.94 1.37 0.52 2.24 1.72 a not calculated.


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Figure 10: TOC graphic


Resonance State Method for Electron Injection in ... - ACS Publications

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