Singlet Excitation Energy Transfer Mediated by Local Exciton Bridges

Article pubs.acs.org/JPCC

Singlet Excitation Energy Transfer Mediated by Local Exciton Bridges Tsutomu Kawatsu,†,‡ Kenji Matsuda,∥ and Jun-ya Hasegawa*,†,§,∥ †

Fukui Institute for Fundamental Chemistry, Kyoto University, 34-4 Takano-nishihiraki-cho, Sakyo-ku, Kyoto 606-8103, Japan Institute for Molecular Science, National Institute of Natural Science, 38 Nishigo-Naka, Myodaiji, Okazaki 444-8585, Japan § Quantum Chemical Research Institute (QCRI), Kyodai Katsura Venture Plaza, North Building 106, 1-36 Goryo-Oohara, Nishikyo-ku, Kyoto, 615-8245 Japan ∥ Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Kyoto-Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan ‡

ABSTRACT: Singlet excitation energy transfer (EET) generally occurs via a Förster mechanism and originates from direct electronic coupling between the donor (D) and acceptor (A). However, indirect EET (the so-called superexchange mechanism) can be crucial in some situations. In this study, we have explored the contributions of various linker types, linker lengths, and tunneling energies in D−linker−A molecular systems using quantum chemical methods [Kawatsu et al. J. Phys. Chem. A 2011, 115, 10814], obtaining conditions in which the superexchange term emerges. To analyze the results in terms of physical parameters, a model Hamiltonian was developed and found to yield results qualitatively consistent with those from the quantum chemical calculations. In a model compound, the superexchange term can be up to 4 times larger than the direct (Förster) term when the energy of the excited state at a linker unit is close to the tunneling energy and when the interactions between these excited states are strong. The ratio of the indirect term to the direct term is greatest at a certain D−A distance because the number of multistep terms increases and the sizes of multistep indirect terms decay exponentially. Single-step indirect terms, which exhibit a polynomial decay, dominate the indirect term for larger D−A distances. The decay behavior with the distance is very different from that of the McConnell type donor−bridge−acceptor superexchange mechanism proposed for the charge transfer (CT). The result of pathway analysis showed that the exciton states mediate EET. The coupling between the local excited states is driven by the pseudo-Coulombic interaction, even in the superexchange terms.

1. INTRODUCTION Excitation energy transfer (EET) is a fundamental process for energy exchange in various systems. An excited state is created by either light absorption or chemical reaction and moves to another excited state in a different molecule or domain via EET. The transferred excitation energy may act as the driving force of luminescence, electron transfer, and other chemical reactions. In biological systems, EET from the light-harvesting antenna to the reaction center creates long-distance charge separation in photosynthetic systems.1−5 Another biological example is EET from aequorin to green fluorescence protein (GFP) in Aequorea victoria,6,7 which has already been applied to luminescence probes.8,9 In the field of engineering, EET plays crucial roles in organic solar cells10,11 and light-emitting diodes.12,13 In EET theory,14−34 the expression for the rate constant is derived from Fermi’s golden rule and includes both electronic coupling and nuclear coupling between the initial and final electronic states. The coupling of the nuclear wave functions determines the density of the contributed electronic states, and several models have been proposed.5,14,18,19,26,34,35 The electronic coupling is often discussed with two different mechanisms originated by the pseudo-Coulombic and exchange interactions. The pseudo-Coulombic interaction, which is © 2012 American Chemical Society

discussed in this paper, was adopted in the Fö r ster mechanism.14 χ dipole = 3 |ma ||mb| Tab (1) R where ma and mb are the transition dipole moments of the initial state a and final state b, respectively. R is the distance between the donor and acceptor. The angular factor χ is χ = cos θab − 3 cos θaR cos θ bR

(2)

θab is the angle between ma and mb, whereas θaR or θbR are the angles between ma or mb and the transition vector from a to b, respectively. To include the exchange, so-called Dexter type interaction,16 exchange integrals must be calculated. These Coulombic and exchange interactions appear together in the electronic coupling formulas for singlet EET when the original Hamiltonian for electrons is used. General quantum chemical computational methods include both of these interactions in the electronic coupling.36 Received: April 22, 2012 Revised: June 5, 2012 Published: June 11, 2012 13865

dx.doi.org/10.1021/jp303878s | J. Phys. Chem. C 2012, 116, 13865−13876

The Journal of Physical Chemistry C

Article

method for numerically classifying the single-step and multistep EET is also given and verified in section 2.3. We introduce a model Hamiltonian approach to extract the essential information from the numerical results (section 2.4), and the parameters used for the model Hamiltonian are derived in section 2.5. In section 3, the results of the calculations are presented and discussed in detail. We describe the results of the quantum chemical calculations for the direct and indirect terms in sections 3.1 and 3.2, respectively. Next, the indirect terms are decomposed into single-step and multistep terms in section 3.3, and the results are further analyzed by the model Hamiltonian in section 3.4. The results of the EET pathway analysis are given in section 3.5. The conclusion of the present study is briefly summarized in section 4.

Electronic excited states other than those of the donor and acceptor can also contribute to the electronic coupling as bridge states of the superexchange mechanism. For example, the following bridge states have been studied: CT states between donor and acceptor molecules,36,37 a ground state coupled to the photonic state,34 and excited states and CT states in the surrounding environment.36 Recently, we have developed a computational method to calculate the bridge-mediated electronic coupling between the donor and acceptor states. We have also developed a method to analyze the EET pathways through the bridge states using the tunneling electronic configuration flux.36 These methods were applied to singlet EET in the model peptide systems.36,38 In these systems, the direct term (the Förster mechanism) dominates the EET mechanism. The bridge-mediated pathways only contributed slightly, while the single-step pathways through the π−π* excited states in the peptide bonds or aromatic amino residues made large contributions. Under what conditions and for which types of systems is bridge-mediated EET significant? In the case of triplet EET, the bridge-mediated superexchange interaction overshadows the direct interaction39−41 because of the small magnitude of the long-range Coulomb mechanism due to the spin forbidden transitions involved. In contrast, because of the Förster mechanism,14 the direct term without the bridge effect is the major pathway in the singlet EET case.42 However, singlet EET can be driven by bridge-mediated superexchange in πconjugated bridged molecules.43−49 Scholes and co-workers have investigated the exciton interaction in a series of naphthalene−bridge−naphthalene (polynorbornyl-bridged dinaphthyl) molecules called DN-2, DN-4, and DN-6 (2, 4, and 6 denote the number of σ-bonds between the naphthalene chromophores).45 These researchers reported that the exciton interactions for DN-4 and DN-6, but not DN-2, are driven by the superexchange mechanism. On the other hand, Pettersson et al. have experimentally and theoretically studied (zincporphyrin)−bridge−porphyrin molecules with zinc centers with various lengths of the bridge.47 They suggested that the ratio between the Förster mechanism and superexchange mechanism is independent of the bridge length but dependent on the type of zinc-porphyrin ligand. Because these two sets of experimental results are inconsistent with each other, theoretically clarifying the conditions in which the bridgemediated EET dominates in the singlet EET is of great interest. In the present study, we calculated the electronic coupling of EET for donor−linker−acceptor molecular systems having different properties. We explored different conditions to enforce the superexchange interaction. First, the tunneling energy was controlled by replacing the donor and acceptor fragments. Second, the type of linker fragment was modified to change the magnitude of the interactions between local bridge states in the linker fragment. Third, the number of linker units in the linker fragment was controlled to change the distance between the donor and acceptor. We also introduced a model Hamiltonian for one-dimensional simple donor−bridge− acceptor models to investigate how these parameters affect the EET electronic coupling. Comparing the results of the model Hamiltonian with those of quantum-chemical calculations, we investigated the adequacy of the simple model and the role and behavior of the parameters in the Hamiltonian. In section 2, we describe theoretical methods for calculating the electronic coupling and tunneling configuration flux (section 2.1) and computational details (section 2.2). A

2. THEORETICAL AND COMPUTATIONAL DETAILS 2.1. Electronic Coupling and Tunneling Configuration Flux. We consider a two-state system for EET in a connected donor−linker−acceptor system. The system is divided into donor, linker, and acceptor fragments. The initial and final states, ΨI and ΨF, are defined as a linear combination of the Slater determinants {Φp}. ΨI =

∑ Cp̅ I(E)Φp (3)

p∉A

ΨF =

∑ Cp̅ F(E)Φp (4)

p∉D

The Slater determinants, {Φp}, are composed of localized molecular orbitals (LMO) that have a population in one of the fragments. “D” and “A” indicate locally excited determinants in the donor and acceptor fragments, respectively. In eqs 3 and 4, the initial and final state wave functions extend to the bridge states via the configuration interaction (CI) but not to A and D, respectively. The tunneling energy, E, is a predefined parameter used to determine the CI coefficients, {C̅ Ip(E)} and {C̅ Fp(E)}. Using eqs 3 and 4, the electronic coupling TIF is written as36,50−54 TIF(E) = ⟨Ψ|I E − Ĥ |ΨF⟩ =

∑ Cp̅ I(E)(E1 − H)pq Cq̅ F(E) pq

(5)

The CI coefficients that appear in eqs 3 and 4 are computed using Green’s function of the perturbation states, Gpq(E)  (1/(H − E1))pq, as55−57 Cp̅ I(E) = −C̅ DI0(E)

∑

HD0pGpq (E) (6)

p ∉ A,p ≠ D0

Cp̅ F(E) = −C̅ AF0(E)

∑

Gqp(E)HpA

p ∉ D,p ≠ A 0

0

(7)

Here, we assumed that determinants D0 and A0 are dominant in D and A, respectively, and that the effect of other excited determinants can be described by a perturbation treatment. The coefficients for the D0 and A0 determinants are determined using the following normalizations.55 |C̅ DI0(E)|2 = 1 −

∑

|Cp̅ I(E)|2 (8)

p ≠ D0

|C̅ AF0(E)|2 = 1 −

∑ p≠A 0

13866

|Cp̅ F(E)|2 (9)



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We divided the electronic coupling TIF into the direct and indirect (superexchange mechanism) terms, yielding the following equations. direct TIF(E) = TIF (E) + TIFindirect(E)

direct TIF (E ) =

∑

(10)

∑

(11)

Cp̅ I(E)(E1 − H)pq Cq̅ F(E)

p ∉ D or q ∉ A

KLM(E) =

(12)

Note the assumptions, C̅ Fp∈D = 0 and C̅ Ip∈A = 0, mentioned above. To investigate the components of TIF, we used the tunneling configuration flux, Jpq,58−61 which was calculated using CI coefficients and Hamiltonian matrix elements as36,62,63 Jpq (E) =

1 I (Cp̅ (E)Cq̅ F(E) − Cp̅ F(E)Cq̅ I(E))Hpq ℏ

ℏJLM (E) TIF(E)

(19)

We thereafter express the E dependence of TIF, JLM, and KLM for convenience. 2.2. Model Systems and Computational Details. To investigate the effects of the tunneling energy E and the electronic structures of the linker, we prepared model compounds with different donor, linker, and acceptor types. To change the tunneling energy E, we selected four donor and acceptor types, naphthalene, anthracene, naphthacene, and pentacene (Figure 1a), with similar electronic structures and

(13)

To understand the flux from a coarse-grained perspective, we accumulated the configuration contributions to define an intergroup tunneling configuration flux between groups L and M. JLM (E) =

(18)

We consider TIF to be the summation of the pathways, such as group D → group L → group M → group A, and investigated each pathway. To compare the results of various model compounds, we normalize JLM as

Cp̅ I(E)(E1 − H)pq Cq̅ F(E)

p ∈ D,q ∈ A

TIFindirect(E) =

∑ L∈X

⎧TIF(E) X = D ⎪ ∑ ℏJLM (E) = ⎨−TIF(E) X = A ⎪ M D, A ∉ X ⎩0

∑ Jpq (E)PpLPqM pq

(14)

Here, a “group” was defined as a group of excitations between two units and a “unit” is a part of the system. PLp in eq 14 is the population of an excited determinant p on the group L. When the group L is defined as excitations from units m to n and the determinant p is an excitation from molecular orbitals (MOs) i to a, PLp is defined as the product of ρ̅na and ρ̅m i .

PpL = ρa̅ n ρi̅ m

(15)

ρ̅na and ρ̅m i are the populations of MOs a and i on units n and m, respectively. D and A are a set of determinants expressing local excitations within the donor and acceptor units, respectively, and are therefore included as a part of the groups. The MO populations ρ̅na and ρ̅m i were determined as

Figure 1. Donor−linker−acceptor molecular model. (a) The four donor and acceptor types. (b) The three linker types (unit length of three).

different excitation energies. We also expected that the donor− acceptor distance is similar for pairs of these compounds. The first excitation energies of these molecules calculated at the CIS/D9565 level are shown in Table 1. Naphthalene showed

⎧ ϑnr

ρr̅ n

where

n ∈ D, A ⎪ = ⎨ ρn / m ⎪ r ∑ ρr else m ∉ D,A ⎩

ϑnr

(16)

Table 1. The First Excitation Energies of the Donor/ Acceptor Fragments Calculated at the CIS/D95 level

is a step function.

⎧1 if r belongs to n ϑnr = ⎨ ⎩ 0 else

(17)

ρ̅an is a raw orbital population obtained using Lö wdin population analysis.64 We treated the donor and acceptor fragments as donor and acceptor units, respectively. To obtain MOs localized within the donor and acceptor units, we applied the minimum orbital deformation (MOD) method.62,63 Reference MOs for the donor and acceptor moieties were calculated in a preliminary step, and the canonical MOs were transformed to maximize the overlap integral with the LMOs. JLM fulfills the following conservation rule. Each JLM looks like a component of pathways from D to A. JLM becomes TIF when L belongs to D or M belongs to A.

donor/acceptor

E (eV)

naphthalene anthracene naphthacene pentacene

5.32 4.30 3.53 2.95

the largest excitation energy (5.32 eV). Extending the πconjugation, the first excitation energy becomes small. The result for pentacene was the smallest (2.95 eV). Considering the EET between the first excited states of the donor and acceptor, the first excitation energy is appropriate for the tunneling energy E. To discuss the bridge effect clearly, the same molecule was used for the donor and acceptor fragments. 13867



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that flow out from L to other bridge groups and by those that flow into L from other groups. Therefore, the amount of the compensation, which is the difference between JDL and JLA, belongs to the multistep pathways. When the direction (sign) of JDL and JLA is the same, the smaller term gives the upper limit of the single-step pathway through L. When the direction of JDL is opposite to that of JAL, the single-step contribution of L is zero. Here, we assume that the flux for the single-step pathway is very close to the upper limit. Under this assumption, the single-step pathway through group L is defined numerically as

To investigate the effects of different donor−acceptor distances, the linker was designed as a one-dimensional chain composed of an oligomer of the same units. The length of the linker ranged from two to six units. Figure 1b shows examples of the anthracene type of the donor and acceptor with three linker units. To change the interactions between the units in the linker, we used a singly bonded (SB) phenylene unit (type 1 in Figure 1b), a doubly bonded (DB) phenylene unit (type 2 in Figure 1b), and a peptide bond unit (type 3 in Figure 1b). The strength of the interaction between the linker units ordered from weakest to strongest is as follows: peptide bond units, phenylene-SB units, and phenylene-DB units. In the present study, we calculated all possible combinations of the donor−linker−acceptor. The total number of model compounds used for the TIF calculations is 60, which is 4 donor/ acceptor types × 5 unit lengths (#Units, from 2 to 6) × 3 linker types. The model compounds were first constructed using ChemDraw,66 and the structures were optimized at the B3LYP67,68/6-31G(d)69 level in the gas phase. The MOs in the donor and acceptor fragments were localized using the MOD62,63 method. The D95 basis set65 was used for all of the calculations. We also used the D95 sets to calculate the reference MOs used in the MOD transformation with the exception of those of the link atoms, which were calculated using the STO-6G set.70 The CIS determinants including these LMOs were used as basis functions to calculate the electronic coupling and the tunneling configuration flux. For the primary donor D0 and acceptor A0 determinants in eqs 6−9, the HOMO−LUMO excited determinant of the donor and acceptor fragments was adopted because of the character of the first excited state of the donor and acceptor fragments. The tunneling energy E was the first excitation energy of the donor and acceptor fragments, as shown in Table 1. Computations were performed using the Gaussian 03 program71 modified for the present purposes. 2.3. Numerical Decomposition of the Intergroup Tunneling Configuration Flux into Single-Step and Multistep Pathways. The EET pathway can be defined by connecting the tunneling configuration fluxes. To investigate the tunneling pathways in more detail, we divided the bridgemediated indirect tunneling pathways into single-step pathways and others, as schematically shown in Figure 2. Here, a single-

single ‐ step JDL/LA

⎧ JDL min(|JDL | , |JLA |) when JDLJLA > 0 ⎪ = ⎨ |JDL | ⎪ when JDLJLA ≤ 0 ⎩0 (20)

where JDL/|JDL| gives the sign of the flux. Positive and negative signs indicate directions from D to A and from A to D, respectively. The total component of TIF that arises from the single-step pathways is defined as single ‐ step TIF ≡ℏ

∑

single ‐ step JDL/LA

L ∉ D,A

(21)

That of the multistep pathways is defined as single ‐ step TIFmultistep ≡ TIFindirect − TIF

(22)

We checked the validity of the above assumption and whether the numerical decomposition worked correctly. As reported in our previous article,36 TIF of the single-step and multistep EETs exhibits polynomial and exponential decays with increasing donor−acceptor distance, respectively. In section 3.4, we also discuss the decay behavior of the singlestep and multistep EET using the model Hamiltonian (see section 2.4) and confirmed that the polynomial and exponential decays occur in the single-step and multistep EET, respectively. The least-squares method was used to fit the following functions to the Tsingle‑step and Tmultistep plots IF IF log10|TIF| = C1 log10 RDA + C2

(23)

log10|TIF| = C1RDA + C2

(24)

C1 and C2 are fitting parameters. The former and latter functions express polynomial and exponential decays with RDA, respectively. The computed fitting parameter of the slope (C1) and its error (standard deviation) for the phenylene-SB and -DB linker systems are shown in Tables 2 and 3, respectively. In all cases, the single-step and multistep components have a small fitting error for eqs 23 and 24, respectively. This good fit indicates that Tsingle‑step and Tmultistep decay polynomially and IF IF exponentially, respectively, with the donor−acceptor distance. This result evaluated the validity of our assumption for the single-step pathway and demonstrated that our numerical dividing method for the single-step and multistep components of TIF works well. An exception is the case of #Unit = 2 in the phenylene-DB linker systems. Because the CT states between two linker units were important bridge states, it was impossible for a two-unit linker to define multistep EET. We therefore used these plots more than those for #Unit ≥ 3 for the fitting procedure. 2.4. Model Hamiltonian Approach for the Indirect Terms. To understand the result of the quantum chemical calculations for the electronic coupling and the flux analysis, we

Figure 2. The intergroup tunneling configuration fluxes of an indirect term are numerically decomposed into single-step and multistep pathways. Arrows indicate the tunneling configuration fluxes. The width of the arrow is proportional to the size of the flux.

step pathway is defined as a connection of two fluxes mediated by a single bridge state, starting from the donor state and arriving at the acceptor state. We define the other pathways as multistep pathways. We focus on the tunneling fluxes between the donor group D and a bridge group L and between L and the acceptor group A. When there are fluxes from D to L (JDL) and from L to A (JLA), the difference between JDL and JLA is compensated by fluxes 13868



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Table 2. The Least Square Fitting of the Single-Step and Multistep Components of TIF at Distances RDA to Polynomial and Exponential Functionsthe Result for the Phenylene-SB Linker System single-step polynomiala

a

multistep exponentialb

polynomiala

exponentialb

phenylene-SB

slope

std. dev.

slope

std. dev.

slope

std. dev.

slope

std. dev.


−3.24(3) −3.10(9) −3.11(6) −3.08(11)

0.0091 0.0265 0.0171 0.0313

−0.067(4) −0.065(3) −0.065(4) −0.065(2)

0.0565 0.0391 0.0484 0.0341

−3.11(21) −2.75(33) −2.89(25) −2.95(31)

0.0608 0.0947 0.0715 0.0895

−0.0653(7) −0.059(3) −0.061(1) −0.063(2)

0.0093 0.0372 0.0181 0.0293

Least square fitting to log10|TDA| = C1 log10 RDA + C2. bLeast square fitting to log10|TIF| = C1RDA + C2.

Table 3. The Least Square Fitting of the Single-Step and Multistep Components of TIF at Distances RDA to Polynomial and Exponential Functionsthe Result for the Phenylene-DB Linker Systema single-step polynomialb

a

multistep exponentialc

polynomialb

exponentialc

phenylene-DB

slope

std. dev.

slope

std. dev.

slope

std. dev.

slope

std. dev.


−3.00(6) −3.04(4) −2.99(7) −3.04(6)

0.0099 0.0064 0.0127 0.0111

−0.062(2) −0.063(3) −0.062(2) −0.063(2)

0.0184 0.0254 0.0157 0.0184

−1.86(15) −2.09(15) −2.07(25) −2.32(21)

0.0253 0.0259 0.0434 0.0360

−0.039(2) −0.044(1) −0.044(3) −0.048(2)

0.0133 0.0084 0.0238 0.0135

The fitting excludes models with #Unit=2. bLeast square fitting to log10|TDA|=C1 log10 RDA + C2. cLeast square fitting to log10|TIF| = C1RDA + C2.

is inversely proportional to the cubed distance.14 The interaction parameters between the donor D and bridge state i, those between the acceptor A and bridge state i, and those between bridge states i and j are written as

also calculated the electronic coupling with simplified numerical models using a parametrized Hamiltonian for the EET in the model compounds. The energy diagram of the model compounds is illustrated in Figure 3a. The horizontal and

⎛ R ⎞−3 βDi = β ⎜ Di ⎟ = βi−3 ⎝ ΔR ⎠

(25)

⎛ R ⎞−3 βi A = β ⎜ i A ⎟ = β(N + 1 − i)−3 ⎝ ΔR ⎠

(26)

⎛ R ij ⎞−3 −3 γij = γ ⎜ ⎟ = γ | i − j| ⎝ ΔR ⎠

(27)

respectively. β and γ without subscripts are constants and denote interactions when the distance is ΔR. The electronic coupling TIF for the fully interacting model (Figure 3a) is expressed using the parameters defined above.

Figure 3. One-dimensional EET models. (a) Model 3a. Original (fully interacted). (b) Model 3b. No unit−unit interactions. (c) Model 3c. Nearest neighbor interaction. (d) Model 3d. Nearest neighbor interaction for the donor and acceptor.

TIF =

vertical axes qualitatively represent a real-space coordinate and the potential energy of local exciton states, respectively. The horizontal bars represent the energy levels of the excited states involved in EET. Because exciton states are the primary intermediate states in our former study,36 we only included exciton states for the bridge states; no CT states were included. The indices D, B, and A indicate the donor, bridge, and acceptor states, respectively. Indices i and j (i, j = 1, ..., N) are for the bridge states in the linker fragment in the order from the donor side to the acceptor sides. The distances between the neighboring donor, linker unit, and acceptor are a constant ΔR, which is common to all four models. The distances between the donor and bridge i, bridge i and the acceptor, and bridges i and j are defined as RDi = i × ΔR, RiA = (N + 1 − i)ΔR, and Rij = |i − j|ΔR, respectively, and are defined by parameters i and j. We assumed that interactions between these exciton states mainly originate from the pseudo-Coulombic interaction, which

⎛

∑ βDi⎜⎝

⎞ 1 ⎟ β H − E1 ⎠ij j A

⎛ ΔE γ ···⎞ 12 ⎜ ⎟ H − E1 = ⎜ γ21 ΔE ···⎟ ⎜ ⎟ ⎝⋮ ⋮ ⋱⎠

(28)

(29)

ΔE is the energy gap between the donor (acceptor) and bridge states. The potential energies of the donor and acceptor states are the same in this model. Multiplying TIF by a constant, γ/β2, we obtain an expression for the unitless electronic coupling T̅ IF. ⎛γ ⎞ TIF̅ = ⎜ 2 ⎟TIF = ⎝β ⎠

⎛ βDi ⎞⎛ ⎞ ⎛ βj A ⎞ γ ⎟ ⎜ ⎟⎜ ⎜ ⎟⎟ ⎝ β ⎠⎝ H − E1 ⎠ij ⎝ β ⎠

∑⎜

(30)

T̅ IF has the same distance dependence as TIF; however, it depends on only two parameters, ΔE/γ and N. 13869



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To investigate the role of the interactions involved in the model compounds, we built three models with limited interactions as follows. We notate the models shown in Figure 3a, b, c, and d as models 3a, 3b, 3c, and 3d, respectively, for convenience. In the first model, the interactions between bridge states were set to zero, as in Figure 3b. γij = 0

(31)

Here, only the single-step coupling, which is mediated by only one bridge state, is allowed between the donor and acceptor. The second model includes only nearest neighbor interaction, as shown in Figure 3c. This model is equivalent to the superexchange theory proposed by McConnell for the intramolecular CT in the donor−(bridge)n−acceptor system.53

⎧γ j = i ± 1 γij = ⎨ ⎩ 0 others

(32)

⎧ β XY = D1 or NA βXY = ⎨ ⎩ 0 others

(33)

Figure 4. Estimated ΔE/γ for the model compounds. (a) Parameter estimation using the Frankel exciton model. (b) Estimated ΔE/γ values. The indices of the horizontal axis are the numbers of benzene rings in the donor/acceptor. For example, “2” denotes naphthalene. Indices 1 and 2 for phenylene linkers indicate that Emonomer and ELMO‑CIS were used for the parameter estimation, respectively.

Only N-step through-bridge coupling is allowed between the donor and acceptor, which is the opposite case from the Figure 3b model. In the last model, in addition to the interactions in the Figure 3c model, virtual exciton states are fully interacted among the bridge states. The donor and acceptor states were allowed to interact with the neighboring bridge states (Figure 3d). This model excludes single-step coupling. Finally, the direct term is also written as eq 34 under the assumptions above. ⎛R ⎞ TDA = T0⎜ DA ⎟ ⎝ ΔR ⎠

the dimer. The γ values calculated using the former and latter procedures were termed as γ1 and γ2, respectively, in Table 4. Table 4. The Excitation Energies of the Monomer and Dimer in the Linker Units and Interbridge State Interaction Energies γ1 and γ2 (See Section 2.4 for Definitions)

−3

= T0(N + 1)−3

(34)

Here, T0 is a constant that is the direct term in the distance ΔR. 2.5. Parameters for the Molecular Linkers Used in the Model Hamiltonian Approach. For the model Hamiltonian given in section 2.4, we estimated the parameter ΔE/γ. ΔE is defined as Emonomer − Etun. The tunneling energy, Etun, is the first excitation energy of the donor and acceptor (Table 1). Emonomer is the excitation energy of a single bridge state that is a local excited state of a linker unit. γ is an interaction energy between the linker units. We examined two procedures for estimating these parameters. The first procedure is to simply use the first excitation energy of benzene (6.31 eV) for Emonomer. In both the phenylene-SB and -DB cases, the linker unit is a benzene molecule. In this procedure, the Emonomer values of the phenylene-SB and -DB linkers were treated as identical despite the differences in bonding between the linker units. We have also developed an alternative procedure, estimating Emonomer using the Hartree−Fock state of the dimer. Using the MOD62,63 method, we transformed the dimer MOs to be very close to the monomer MOs. Next, a CIS calculation was performed with the active MOs localized in a monomer moiety. The calculated excitation energy, ELMO‑CIS, was used as Emonomer. The monomer energies of the phenylene-SB and -DB linkers differ. We estimated the interaction parameter γ between the bridge states assuming a Frankel exciton15 model, as shown in Figure 4a. The interaction γ was calculated as γ = Emonomer − Edimer or γ = ELMO‑CIS − Edimer, where Edimer is the first excitation energy of

bridge type

Edimer (eV)

Emonomer (eV)

γ1 (eV)

phenylene-SB phenylene-DB peptide

5.76 4.04 9.89

6.31 6.31 8.87

0.53 2.27

ELMO‑CIS (eV)

γ2 (eV)

6.61 6.85 10.05/10.09

0.85 2.81 0.18

For the peptide linker, we assumed that the linker unit is a peptide bond, OCNH. The first excited state of the peptide unit is of n−π* character. The calculated interaction between the n−π* states of the neighbor units was so small that this n−π* state does not work as an effective bridge state. Alternatively, we chose the second excited state, π−π* state, as a bridge state. The π−π* excitation energy was used for ELMO‑CIS, and that of the corresponding state in the peptide dimer was used for Edimer. In the actual calculations, we adopted CH3CONHCH2CONHCH3 for the peptide dimer. Because of the asymmetry in the dimer structure, ELMO‑CIS for the left and the right CONH units were different. Therefore, we averaged the left and right results for ELMO‑CIS in estimating γ. In the calculation, Emonomer is smaller than Edimer due to bonding, and γ1 is not evaluated for this linker. The computed parameters, Edimer, Emonomer, ELMO‑CIS, γ1, and γ2, are shown in Table 4. The calculated γ value for the phenylene-DB unit was significantly large, while that of the peptide unit was smallest. In Figure 4b, the calculated ΔE/γ value was compared among the donor/acceptors for each linker unit. The ΔE/γ values for the peptide linker units were much 13870



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acceptor units are parallel to the donor−acceptor direction. In the models of the phenylene-DB linker, the angular factor χ is 1.6−1.8, and the linearity is worse than that in the SB case because the linker units cannot twist. For the peptide linkers, the χ value depends on the linker length because of the softness of the peptide. 3.2. Direct Term vs Indirect Term. The ratio of the direct and indirect terms was evaluated. In Figure 6, the ratios of the computed direct terms in total electronic coupling TIF were shown for all of the models. In phenylene linkers (SB and DB), the ratio of the indirect terms decreases in the order of the size of the donor and acceptor units (naphthalene > anthracene > naphthacene > pentacene). As shown in Table 1, a smaller molecule has a higher first excitation energy. On the other hand, the first excitation energy of a linker unit of the phenylene, benzene, was calculated to be 6.31 eV (at the CIS/ D95 level) and was larger than that of naphthalene. This finding indicates that the ratio of the indirect term increases when the energy gap between the donor/acceptor states and bridge state decreases. Interestingly, the ratio of the direct term of phenylene-SB has a minimum at RDA from 20 to 25 Å (Figure 6a), which means that the ratio of the indirect term has a maximum in this region. In contrast, the ratio of the indirect term for phenylene-DB monotonically decreased with increasing linker lengths. These behaviors originate from the balance between the two distance dependencies. At longer distances, the indirect term decays faster than the direct term in the donor−acceptor distance.36 At shorter cases, the number of components in the indirect term decreases because fewer bridge states exist. This situation is similar to the balance between the ratio of single-step and multistep pathways, which is discussed later in section 3.4. In these calculations, the highest ratio of the indirect term exceeded 75%. This result indicates that the Fö r ster mechanism14 (eq 1) significantly underestimates TIF, and the error increases to 400% compared to that computed using a quantum chemical approach (eq 5). The total electronic couplings TIF for phenylene-SB and -DB are shown in Figure 7a and b, respectively. The vertical and horizontal axes are in a logarithmic scale. The plots for the direct terms (dashed lines) gave straight lines, indicating that TIF shows a polynomial decay. In contrast, the plots for the total TIF are slightly

larger than those for the phenylene-SB and -DB linker units because of the large ΔE values and the small γ value. Phenylene-SB linker systems have larger ΔE/γ than phenylene-DB linker systems because of the γ value. We also find that a larger donor and acceptor have a larger ΔE/γ value because Etun deceases.

3. RESULTS AND DISCUSSION 3.1. Direct Term Represented by a Pseudo-Coulombic Description. In Figure 5, we compare the numerical result of

Figure 5. Comparison of TIF obtained by the direct term and by the dipole approximation.

the direct term (eq 11) with that obtained with the dipole approximation (eq 1) for all of the molecular models. The center of mass of the carbon atoms was adopted as that of the donor and acceptor in eq 1. As shown in Figure 5, the results of the two calculations agree well, indicating that the direct term originates from the pseudo-Coulombic interaction, as assumed when eq 1 was derived. In other words, the exchange interaction is only a small part of the direct term. In the phenylene-SB linker models, the axis from the donor to acceptor units was fixed for all models and the orientation angler factor χ (eq 2) was 2.0. These settings were chosen because the units in the phenylene linker align linearly and the transition moments of the first excited states in the donor and

Figure 6. The ratio of the direct term in total electronic coupling TIF. (a) Phenylene-SB linker. (b) Phenylene-DB linker. (c) Peptide linker. Squares with numbers indicate molecules with the same number of linker units (#Unit). 13871



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for the three types of linkers. Within the range of RDA, the ratio of the single-step EET was over 50%. The result clearly shows that single-step EET is the major component of the indirect terms in the bridge mediated EET. In phenylene-SB with naphthalene as the donor and acceptor, the indirect term was the major component. Within 2 ≤ #Units ≤ 6, the indirect term comprised more than 60% of the total TIF, as shown in Figure 6a. Over 42% of TIF of the compound originates from the single-step terms, less than 40% from the direct term, and the rest (approximately 20%) from the multistep terms. The importance of the multistep EET increases in the phenylene-DB case. For example, in the case of #Units = 6 and naphthalene as the donor and acceptor, the direct term was just 25% of the total TIF, and approximately 42% was from the single-step EET. The remaining 33% arose from the multistep EET. These particular cases show that the single-step EET is the major component even in singlet EET and that the multistep and single-step EETs can be comparable. In the case of the peptide linker, the single-step ratio is generally high comparing with the multistep, as seen in Figure 8c. However, because the direct term was very dominant in the total TIF, the single-step term was a minor contributor in this series of compounds. The only exception was the case for #Units = 5 with a naphthacene donor/acceptor. The calculated direct term was small because of the orientation factor χ in the computational model. In this case, the single-step term shared 61% of the total TIF, while the direct term shared 30%. We note that the effects from the donor−acceptor orientation did not clearly appear in the indirect terms, which is probably because the summation of many indirect terms averaged the variation in the numerical results. We note that the definitions of the single-step and multistep pathways depend on the definitions of the bridge fragments. We adopted the present decomposition to compare the phenylene-SB result with the DB one because we are interested in how the interactions between the fragments affect the mechanism. If a fragment is divided into small pieces of fragments, the tunneling configuration fluxes are also divided, and some of the single-step pathways become multistep ones. Consequently, the ratio of the multistep pathways against the single-step ones increases. However, such multistep pathways run only the local region of the bridge, and a “single-step”-like view is unchanged. In addition, the present definition of the

Figure 7. Computed total electronic coupling (TIF). Comparison of total TIF with the direct term. (a) Phenylene-SB and (b) phenyleneDB. The normal and dashed lines represent total TIF and the direct terms, respectively.

rounded, indicating the effect of the indirect term (clearest for the naphthalene donor/acceptor case). In the peptide linker system, the ratio changed discontinuously, as shown in Figure 6c, because the orientation angler factor χ varies due to the donor−acceptor orientation. This discontinuity covers systematic behaviors observed in the phenylene cases, such as the distance RDA and the choice of the donor and acceptor. In general, these systems have a greater weight in the direct term than the phenylene linker systems. An exceptional case is a compound with naphthacene donor/ acceptor and #Unit = 5. The electronic coupling is dominated by the indirect term because of the small χ value. These examples show that, even in singlet EET, a condition exists in which the indirect EET dominates the direct EET even though most of the interactions between two states are driven by the Förster type pseudo-Coulombic interaction. 3.3. Single-Step vs Multistep in the Indirect Term. To investigate components of the indirect term, we divided the computed indirect term into single-step and multistep components using the numerical decomposition method described in section 2.3. Here, we discuss the distance dependence of the single-step ratio in the indirect term (=single-step + multistep). In Figure 8, the ratios were plotted

Figure 8. The distance dependence of the ratio of the single-step component in the indirect term of TIF. (a) Phenylene-SB linker systems. (b) Phenylene-DB linker systems. (c) Peptide linker systems. Squares with numbers indicate molecules having the same number of linker units (#Unit). 13872



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fragment is reasonable because, in the phenylene-DB, the multistep term decays exponentially, as shown in Table 3. We also found that the single-step/multistep ratios in Figure 8 show a similar trend to those between the direct and indirect terms in Figure 6. In particular, the behaviors of the plots in Figure 8a and b are similar to those in Figure 6a and b, respectively. For the phenylene-SB case, the curves are convex downward. For the phenylene-DB case, the curves decrease monotonically. These behaviors depend on the multistep contribution, which will be analyzed using the model Hamiltonian in the next section. 3.4. A Model Hamiltonian Study for the Indirect EET. The quantum chemical calculations for the model compounds considered all possible bridge states and interactions within a given theoretical model. However, because of this generality and complexity, the result is not easily interpreted in terms of physical parameters such as energy differences and interactions between the units. To obtain a clear view of the indirect EET, we introduced a model Hamiltonian including simple physical parameters as described in section 2.4. Because T̅ IF depends only on two parameters, ΔE/γ and N (the number of bridge units), we can explore these parameters in the model calculations. We are particularly interested in how the distance dependence of T̅ IF (eq 30, the distance-dependent factor of TIF) behaves under the variation of the EET models (from 3a to 3d in Figure 3) and the parameters (ΔE/γ and N). This information could also help us understand the results for various donor/acceptors and linker units. First, we used the “fully interacted” model 3a (Figure 3a) to investigate how the T̅ IF value decays with the number of the linker units (from N = 0 to 100). The results of the calculations with three different ΔE/γ values (1.85, 2.00, and 5.00) are shown in Figure 9a. For a larger ΔE/γ value, such as ΔE/γ = 5.0, the T̅ IF decay line is straight. In contrast, when ΔE/γ decreases, the decay line became rounded in the region from N = 3 to 50 (log10 N = 0.5−1.7). The superexchange mechanism frequently breaks down for much smaller ΔE/γ values caused by the resonance between the bridge energy and the tunneling energy. To understand the origin of the decay behaviors, we introduced models 3b, 3c, and 3d, as explained in section 2.4. The ΔE/γ value of 2.0 was used. In model 3b, the indirect EET was limited to single-step EET. As shown in Figure 9b, model 3b produced a decay “line”, indicating that the single-step EET gives a polynomial decay. From the slope of the decay line, the T̅ IF value is proportional to N−3. This result is explained by each state-to-state interaction being proportional to N−3 in the present model. Model 3b is a limit of the weak interunit interaction. In model 3c (McConnell type superexchange regime), only nearest neighbor interactions were allowed in the EET. Thus, only (N + 1)-step (multistep) EET was involved. The result in Figure 9b shows a rounded decay, which is very similar to the rounded decay seen in model 3a with smaller ΔE/γ values. This exponential decay is characteristic of multistep EET, as we described in a previous study.36 Therefore, the rounded decay curves in model 3a originate from the multistep EET. In model 3d, we only allowed the donor and acceptor to interact with their neighbor linker unit; no single-step EET was involved. As seen in Figure 9b, the decay curve for shorter distances was dominated by the exponential type characteristic of the multistep EET, and the curve is very similar to that of the full model 3a. The multistep exponential decay is caused by the increase in the number of

Figure 9. The distance factor T̅ IF (eq 25) for indirect EET versus the number of the bridge units (N). The donor−acceptor distance is defined as (N + 1)ΔR. (a) ΔE/γ dependence. (b) Model dependence at ΔE/γ = 2.0.

the steps, Nstep. On the other hand, the decay of the multistep coupling also includes a polynomial decay. This polynomial decay is because the distance between the donor and acceptor increases under the same Nstep. Such a fixed-Nstep effect resulted in a polynomial decay in the T̅ IF plot even for large N, indicating the contribution of some lower-order multistep EET. The difference between models 3a and 3d in the region of N > 20 shows the approximate single-step EET contribution to the total T̅ IF. In the popular McConnell type superexchange53 model designed for CT, Nstep always is equal to N + 1 and increases with the distance. The TIF value, therefore, decays exponentially (see model 3c in Figure 9b). However, in EET, the couplings with Nstep ≠ (N + 1) dominate and the TIF decays are polynomially caused by the fixed-Nstep effect. Considering the behaviors of models 3b−3d, the N dependence of T̅ IF in model 3a (the full model) arises from the superposition of the exponential and polynomial decays. The former originates from multistep pathways that are enhanced when the number of steps increases. The multistep EET was the predominant portion of the indirect term for intermediate N values. The latter comes mainly from the singlestep pathways at increasing RDi and RiA distances. Because the exponential terms decay faster than the polynomial terms in the region of large N, the rounded decay curves shown in Figure 8a revert to the straight form at a certain N value. We note that in 13873



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Figure 10. Major EET tunneling pathways in the phenylene linker models. (a) The K̃ (z) plot. See text. Major tunneling fluxes for the (b) phenyleneSB and (c) phenylene-DB linkers (|KLM| > 0.01). Blue, red, and orange arrows denote the direct, through-exciton, and through-neighboring-CT fluxes, respectively. Green arrows are through-CT (from linker units i to iii) fluxes. Exciton states are shown using index “ex” with red circles. In particular, D and A states are represented with red ovals. CT states are indicated by the “+ −” index inside yellow ovals. The relative positions of + and − indicate the direction of the CT. Orange and green CT colors indicate the neighboring CT and distant CT, respectively. The width of the arrow is proportional to the magnitude of KLM.

the limit of small N, the number of bridge states is insufficient to enhance the multistep contribution. On the basis of the results of the model Hamiltonian, the numerical behavior seen in Figures 6 and 8 is discussed. In the result for the phenylene-SB linker, both the single-/multistep ratio and the direct/total ratio decreased at intermediate linker lengths because the multistep components have a significant contribution in that region. In the results of the phenylene-DB linker, both the single-/multistep ratio and the direct/total ratio decayed monotonically with increasing linker length. Because the ΔE/γ value of the phenylene-DB linker is smaller than that of phenylene-SB, the multistep region extends to a greater linker length region and the ratio of the multistep EET decreases much more slowly than in the case of phenylene-SB, as shown in Figure 9a. 3.5. EET Pathways. At the end of this paper, we mention the result of the EET pathway analysis using the tunneling configuration flux. Here, we return to the original Hamiltonian and calculate the intergroup tunneling configuration flux for the phenylene linker (SB and DB) systems (#Units = 3) with an anthracene donor/acceptor. For convenience, the three linker units shown in Figure 10b and c were labeled as i, ii, and iii from left to right in the figure. We first show that the major bridge states are exciton groups (intraunit excited states) and CT groups (interunit CT excited states) between neighboring units. Other CT states were neglected from the flux for the present purpose. These five exciton and eight CT groups were labeled 1 for the D → D excitation, 3 for D → i, 11 for i → D, 13 for i → i, 14 for i → ii, 18 for ii → i, 19 for ii → ii, 20 for ii → iii, 24 for iii → ii, 25 for iii → iii, 22 for iii → A, 10 for A → iii, and 7 for A → A. These labels indicate the approximate position of the excitations in the molecule. We next defined a coordinate z. These 12 excitation groups were aligned along the z axis in the above order, as shown in Figure 10a. The z value was incremented between neighboring two groups. The donor group 1 and acceptor group 7 are located at the left and right

edges of the z axis, respectively. If all of the groups in the system are between the donor and acceptor on the z axis, at any z values, the summation of the tunneling fluxes KLM from L (the left side of z) to M (the right side of z) becomes 1 because of eqs 18 and 19. We then define a partial summation of the tunneling flux among the 12 groups. exciton neighboring CT

K̃ (z) ≡

∑ L ∈ left of z M ∈ right of z

KLM (35)

At a specific z value in the one-dimensionally aligned groups, K̃ (z) is defined as the summation of the tunneling configuration flux from a group in the left-hand side of z to a group in the right-hand side of z. K̃ (z) represents a tunneling configuration flux that passes a border defined by z. The computed K̃ (z) is shown in Figure 10a. Although only exciton and neighboring CT groups were considered, the K̃ (z) values for the SB and DB cases were more than 95 and 90%, respectively, and close to 1 for any z value. The result suggested that the exciton and neighboring CT groups are important bridge states. In addition, the direct term contributes only 49 and 40% in the SB and DB cases, respectively, as shown in Figure 6. Within the indirect term, the single-step pathway was important, as shown in Figure 8, and we can conclude that the dominant pathways in the bridge-mediated EET are direct and single-step EETs via the exciton states and the neighboring CT states. The main intergroup tunneling configuration fluxes with | KLM| > 0.01 for the phenylene linkers (SB and DB) are drawn using arrows in Figure 10b and c, respectively. The width of the arrow is proportional to the amount of the tunneling configuration flux. The direct flux (blue arrow) is predominant in all fluxes. The red arrows are through-exciton fluxes and include all pure through-exciton pathways. Exciton states are shown as the “ex” index in the figures. The orange arrows pass 13874



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polynomial way, which is different from the exponential decay seen in the McConnell type superexchange model. The EET pathways were also analyzed. In phenylene linker systems, bridge-mediated EET pathways are single-step mostly either through-exciton or through-neighboring CT states. The exciton states (local excited states in the linker units) are dominant in EET, especially when the interactions between the bridge states are small.

through the neighboring CT groups. CT states are shown using the “+ −” index in the figures, and relative positions of + and − indicate the direction of CT separation. We see the relatively large single-step pathway fluxes: In both SB and DB, the arrows start from D, pass one of the linker units, and finish at D. The phenylene-SB linker has no other dominant pathway, as shown in Figure 10b. In contrast, for the phenylene-DB linker, a CT state between distant units (see green arrows in Figure 10c) also contributed, which represents the difference of the interunit interactions between the phenylene-SB and -DB cases. Calculated tunneling pathways have their own directions, as shown in Figure 10b and c. The directions of the tunneling pathways (configuration fluxes) are related to the phases and determine the interference among the tunneling pathways. This was discussed for the electron transfer.60,72−76 The tunneling pathways with an opposite direction destructively interfere and decrease the total TIF. In contrast, one with the same direction constructively interferes and increases the total TIF. In Figure 10b and c, all of the fluxes run from the donor to acceptor sides, and these tunneling pathways enhance the total TIF. In our previous studies,36,38 destructive interference was observed in an EET mechanism.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81-75-7117867. Fax: +81-75-781-4757. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This study was supported by KAKENHI (No. 21685002) from the Japan Society for the Promotion of Science (JSPS), JSTCREST, and Strategic Programs for Innovative Research (SPIRE). Some of the computations were performed at RCCS (Okazaki, Japan) and ACCMS (Kyoto University).

■

4. CONCLUSIONS In general, singlet EET has been explained by a Förster mechanism that includes a direct electronic coupling between the donor (D) and acceptor (A) states. Considering the longrange EET observed in biology and nanotechnology, the possibility of bridge-mediated indirect EET, so-called superexchange EET, is of particular interest and is well suited for theoretical studies. In this study, we have explored conditions in which the superexchange term in EET dominates and systematically examined various donor/acceptor, linker types, and linker lengths in the donor−linker−acceptor molecular models using the quantum chemical approach developed in our previous study.36 The indirect term in the electronic coupling becomes maximal at a particular distance caused by the following two effects. Although the number of the indirect terms increases with the distance, the indirect terms decay more rapidly than the direct terms at longer distances. In the phenylene-DB case, the indirect term became up to 4 times larger than the direct term. We numerically decomposed the indirect term into the single-step and multistep components. In either very small or large D−A distances, the single-step term was more dominant than the multistep term. The multistep term became dominant only in intermediate distances because of the exponential decay behavior. We also developed a one-dimensional simple model for the bridge-mediated singlet EET for comparison with the results of the quantum chemical models. When the potential energies of the local bridge states are close to the tunneling energy and the interactions between the bridge states are strong enough, the indirect ratio exceeds the direct ratio. The qualitative behavior obtained by quantum chemical calculations was reproduced by the simple model Hamiltonian, in which we assumed only local exciton-like bridge states and pseudo-Coulombic interactions among donor, acceptor, and bridge states. Although the nearest neighbor interaction model is known as McConnell type superexchange for the CT, single-step interactions are important for the singlet EET. Therefore, the TIF for singlet EET of the superexchange type decays dominantly in a

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