Singlet Excitation Energy Transfer Mediated by Local Exciton Bridges

Singlet Excitation Energy Transfer Mediated by Local Exciton BridgesTsutomu 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) generallyoccurs via a Forster mechanism and originates from directelectronic 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 exploredthe contributions of various linker types, linker lengths, andtunneling energies in D−linker−A molecular systems usingquantum chemical methods [Kawatsu et al. J. Phys. Chem. A2011, 115, 10814], obtaining conditions in which the super-exchange term emerges. To analyze the results in terms of physicalparameters, a model Hamiltonian was developed and found to yield results qualitatively consistent with those from the quantumchemical calculations. In a model compound, the superexchange term can be up to 4 times larger than the direct (Forster) termwhen the energy of the excited state at a linker unit is close to the tunneling energy and when the interactions between theseexcited states are strong. The ratio of the indirect term to the direct term is greatest at a certain D−A distance because thenumber 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 isvery different from that of the McConnell type donor−bridge−acceptor superexchange mechanism proposed for the chargetransfer (CT). The result of pathway analysis showed that the exciton states mediate EET. The coupling between the localexcited states is driven by the pseudo-Coulombic interaction, even in the superexchange terms.

1. INTRODUCTIONExcitation energy transfer (EET) is a fundamental process forenergy exchange in various systems. An excited state is createdby either light absorption or chemical reaction and moves toanother excited state in a different molecule or domain via EET.The transferred excitation energy may act as the driving force ofluminescence, electron transfer, and other chemical reactions.In biological systems, EET from the light-harvesting antenna tothe reaction center creates long-distance charge separation inphotosynthetic systems.1−5 Another biological example is EETfrom aequorin to green fluorescence protein (GFP) in Aequoreavictoria,6,7 which has already been applied to luminescenceprobes.8,9 In the field of engineering, EET plays crucial roles inorganic solar cells10,11 and light-emitting diodes.12,13

In EET theory,14−34 the expression for the rate constant isderived from Fermi’s golden rule and includes both electroniccoupling and nuclear coupling between the initial and finalelectronic states. The coupling of the nuclear wave functionsdetermines the density of the contributed electronic states, andseveral models have been proposed.5,14,18,19,26,34,35 Theelectronic coupling is often discussed with two differentmechanisms originated by the pseudo-Coulombic and exchangeinteractions. The pseudo-Coulombic interaction, which is

discussed in this paper, was adopted in the Forstermechanism.14

χ= | || |TR

m mabdipole

3 a b (1)

where ma and mb are the transition dipole moments of theinitial state a and final state b, respectively. R is the distancebetween the donor and acceptor. The angular factor χ is

χ θ θ θ= −cos 3 cos cosab aR bR (2)

θab is the angle between ma and mb, whereas θaR or θbR are theangles between ma or mb and the transition vector from a to b,respectively. To include the exchange, so-called Dexter typeinteraction,16 exchange integrals must be calculated. TheseCoulombic and exchange interactions appear together in theelectronic coupling formulas for singlet EET when the originalHamiltonian for electrons is used. General quantum chemicalcomputational methods include both of these interactions inthe electronic coupling.36

Received: April 22, 2012Revised: June 5, 2012Published: June 11, 2012

Article

pubs.acs.org/JPCC

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

pubs.acs.org/JPCC

Electronic excited states other than those of the donor andacceptor can also contribute to the electronic coupling as bridgestates of the superexchange mechanism. For example, thefollowing bridge states have been studied: CT states betweendonor and acceptor molecules,36,37 a ground state coupled tothe photonic state,34 and excited states and CT states in thesurrounding environment.36

Recently, we have developed a computational method tocalculate the bridge-mediated electronic coupling between thedonor and acceptor states. We have also developed a method toanalyze the EET pathways through the bridge states using thetunneling electronic configuration flux.36 These methods wereapplied to singlet EET in the model peptide systems.36,38 Inthese systems, the direct term (the Forster mechanism)dominates the EET mechanism. The bridge-mediated pathwaysonly contributed slightly, while the single-step pathwaysthrough the π−π* excited states in the peptide bonds oraromatic 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, thebridge-mediated superexchange interaction overshadows thedirect interaction39−41 because of the small magnitude of thelong-range Coulomb mechanism due to the spin forbiddentransitions involved. In contrast, because of the Forstermechanism,14 the direct term without the bridge effect is themajor pathway in the singlet EET case.42 However, singlet EETcan be driven by bridge-mediated superexchange in π-conjugated bridged molecules.43−49 Scholes and co-workershave investigated the exciton interaction in a series ofnaphthalene−bridge−naphthalene (polynorbornyl-bridged di-naphthyl) molecules called DN-2, DN-4, and DN-6 (2, 4, and 6denote the number of σ-bonds between the naphthalenechromophores).45 These researchers reported that the excitoninteractions for DN-4 and DN-6, but not DN-2, are driven bythe superexchange mechanism. On the other hand, Petterssonet al. have experimentally and theoretically studied (zinc-porphyrin)−bridge−porphyrin molecules with zinc centerswith various lengths of the bridge.47 They suggested that theratio between the Forster mechanism and superexchangemechanism is independent of the bridge length but dependenton the type of zinc-porphyrin ligand. Because these two sets ofexperimental results are inconsistent with each other,theoretically clarifying the conditions in which the bridge-mediated 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 havingdifferent properties. We explored different conditions toenforce the superexchange interaction. First, the tunnelingenergy was controlled by replacing the donor and acceptorfragments. Second, the type of linker fragment was modified tochange the magnitude of the interactions between local bridgestates in the linker fragment. Third, the number of linker unitsin the linker fragment was controlled to change the distancebetween the donor and acceptor. We also introduced a modelHamiltonian for one-dimensional simple donor−bridge−acceptor models to investigate how these parameters affectthe EET electronic coupling. Comparing the results of themodel Hamiltonian with those of quantum-chemical calcu-lations, we investigated the adequacy of the simple model andthe 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

method for numerically classifying the single-step and multistepEET is also given and verified in section 2.3. We introduce amodel Hamiltonian approach to extract the essentialinformation from the numerical results (section 2.4), and theparameters used for the model Hamiltonian are derived insection 2.5. In section 3, the results of the calculations arepresented and discussed in detail. We describe the results of thequantum chemical calculations for the direct and indirect termsin sections 3.1 and 3.2, respectively. Next, the indirect terms aredecomposed into single-step and multistep terms in section 3.3,and the results are further analyzed by the model Hamiltonianin section 3.4. The results of the EET pathway analysis aregiven in section 3.5. The conclusion of the present study isbriefly summarized in section 4.

2. THEORETICAL AND COMPUTATIONAL DETAILS2.1. Electronic Coupling and Tunneling Configuration

Flux. We consider a two-state system for EET in a connecteddonor−linker−acceptor system. The system is divided intodonor, linker, and acceptor fragments. The initial and finalstates, ΨI and ΨF, are defined as a linear combination of theSlater determinants {Φp}.

∑Ψ = Φ∉

C E( )Ip A

pI

p(3)

∑Ψ = Φ∉

C E( )Fp D

pF

p(4)

The Slater determinants, {Φp}, are composed of localizedmolecular orbitals (LMO) that have a population in one of thefragments. “D” and “A” indicate locally excited determinants inthe donor and acceptor fragments, respectively. In eqs 3 and 4,the initial and final state wave functions extend to the bridgestates via the configuration interaction (CI) but not to A and D,respectively. The tunneling energy, E, is a predefined parameterused to determine the CI coefficients, {Cp

I (E)} and {CpF(E)}.

Using eqs 3 and 4, the electronic coupling TIF is writtenas36,50−54

∑= ⟨Ψ| − |Ψ ⟩ = − T E E H C C E1 H( ) (E)(E ) ( )IF I Fpq

pI

pq qF

(5)

The CI coefficients that appear in eqs 3 and 4 are computedusing Green’s function of the perturbation states, Gpq(E) (1/(H − E1))pq, as

55−57

∑ = − ∉ ≠

C E C E H G E( ) ( ) ( )pI

DI

p A,p DD p pq0

0

0(6)

∑ = − ∉ ≠

C E C E G E H( ) ( ) ( )pF

AF

p D,p Aqp pA0

0

0(7)

Here, we assumed that determinants D0 and A0 are dominant inD and A, respectively, and that the effect of other exciteddeterminants can be described by a perturbation treatment.The coefficients for the D0 and A0 determinants are determinedusing the following normalizations.55

∑| | = − | |≠

C E C E( ) 1 ( )DI 2

p DpI 2

0

0 (8)

∑| | = − | |≠

C E C E( ) 1 ( )AF 2

p ApF 2

0

0 (9)

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp303878s | J. Phys. Chem. C 2012, 116, 13865−1387613866

We divided the electronic coupling TIF into the direct andindirect (superexchange mechanism) terms, yielding thefollowing equations.

= +T E T E T E( ) ( ) ( )IF IFdirect

IFindirect

(10)

∑= − ∈ ∈

T E C E C E1 H( ) ( )(E ) ( )IFdirect

p D,q ApI

pq qF

(11)

∑= − ∉ ∉

T E C E C E1 H( ) ( )(E ) ( )IFindirect

p D or q ApI

pq qF

(12)

Note the assumptions, Cp∈DF = 0 and Cp∈A

I = 0, mentionedabove.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

=ℏ

− J E C E C E C E C E H( )1

( ( ) ( ) ( ) ( ))pq pI

qF

pF

qI

pq (13)

To understand the flux from a coarse-grained perspective, weaccumulated the configuration contributions to define anintergroup tunneling configuration flux between groups L andM.

∑=J E J E P P( ) ( )LMpq

pq pL

qM

(14)

Here, a “group” was defined as a group of excitations betweentwo units and a “unit” is a part of the system. Pp

L in eq 14 is thepopulation of an excited determinant p on the group L. Whenthe group L is defined as excitations from units m to n and thedeterminant p is an excitation from molecular orbitals (MOs) ito a, Pp

L is defined as the product of ρan and ρi

m.

ρ ρ= PpL

an

im

(15)

ρan and ρi

m are the populations of MOs a and i on units n and m,respectively. D and A are a set of determinants expressing localexcitations within the donor and acceptor units, respectively,and are therefore included as a part of the groups.The MO populations ρa

n and ρim were determined as

∑ρ ρ ρ =ϑ ∈

∉

⎧⎨⎪

⎩⎪n D, A

/ elsern

rn

rn

m D,Arm

(16)

where ϑrn is a step function.

ϑ =⎧⎨⎩

1 if r belongs to n

0 elsern

(17)

ρan is a raw orbital population obtained using Lowdin

population analysis.64 We treated the donor and acceptorfragments as donor and acceptor units, respectively. To obtainMOs localized within the donor and acceptor units, we appliedthe minimum orbital deformation (MOD) method.62,63

Reference MOs for the donor and acceptor moieties werecalculated in a preliminary step, and the canonical MOs weretransformed 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 TIFwhen L belongs to D or M belongs to A.

∑ ∑ ℏ =

=

− =

∉∈

⎧⎨⎪

⎩⎪J E

T E

T E( )

( ) X D

( ) X A

0 D, A XL X M

LM

IF

IF

(18)

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

=ℏ

K EJ E

T E( )

( )

( )LMLM

IF (19)

We thereafter express the E dependence of TIF, JLM, and KLMfor convenience.

2.2. Model Systems and Computational Details. Toinvestigate the effects of the tunneling energy E and theelectronic structures of the linker, we prepared modelcompounds with different donor, linker, and acceptor types.To change the tunneling energy E, we selected four donor andacceptor types, naphthalene, anthracene, naphthacene, andpentacene (Figure 1a), with similar electronic structures and

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

the largest excitation energy (5.32 eV). Extending the π-conjugation, the first excitation energy becomes small. Theresult for pentacene was the smallest (2.95 eV). Consideringthe EET between the first excited states of the donor andacceptor, the first excitation energy is appropriate for thetunneling energy E. To discuss the bridge effect clearly, thesame molecule was used for the donor and acceptor fragments.

Figure 1. Donor−linker−acceptor molecular model. (a) The fourdonor and acceptor types. (b) The three linker types (unit length ofthree).

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

donor/acceptor E (eV)

naphthalene 5.32anthracene 4.30naphthacene 3.53pentacene 2.95



To investigate the effects of different donor−acceptordistances, the linker was designed as a one-dimensional chaincomposed of an oligomer of the same units. The length of thelinker ranged from two to six units. Figure 1b shows examplesof the anthracene type of the donor and acceptor with threelinker units. To change the interactions between the units inthe linker, we used a singly bonded (SB) phenylene unit (type1 in Figure 1b), a doubly bonded (DB) phenylene unit (type 2in Figure 1b), and a peptide bond unit (type 3 in Figure 1b).The strength of the interaction between the linker unitsordered from weakest to strongest is as follows: peptide bondunits, phenylene-SB units, and phenylene-DB units. In thepresent study, we calculated all possible combinations of thedonor−linker−acceptor. The total number of model com-pounds used for the TIF calculations is 60, which is 4 donor/acceptor types × 5 unit lengths (#Units, from 2 to 6) × 3 linkertypes.The model compounds were first constructed using

ChemDraw,66 and the structures were optimized at theB3LYP67,68/6-31G(d)69 level in the gas phase. The MOs inthe donor and acceptor fragments were localized using theMOD62,63 method. The D95 basis set65 was used for all of thecalculations. We also used the D95 sets to calculate thereference MOs used in the MOD transformation with theexception of those of the link atoms, which were calculatedusing the STO-6G set.70 The CIS determinants including theseLMOs were used as basis functions to calculate the electroniccoupling and the tunneling configuration flux. For the primarydonor D0 and acceptor A0 determinants in eqs 6−9, theHOMO−LUMO excited determinant of the donor andacceptor fragments was adopted because of the character ofthe first excited state of the donor and acceptor fragments. Thetunneling energy E was the first excitation energy of the donorand acceptor fragments, as shown in Table 1. Computationswere performed using the Gaussian 03 program71 modified forthe present purposes.2.3. Numerical Decomposition of the Intergroup

Tunneling Configuration Flux into Single-Step andMultistep Pathways. The EET pathway can be defined byconnecting the tunneling configuration fluxes. To investigatethe tunneling pathways in more detail, we divided the bridge-mediated indirect tunneling pathways into single-step pathwaysand others, as schematically shown in Figure 2. Here, a single-

step pathway is defined as a connection of two fluxes mediatedby a single bridge state, starting from the donor state andarriving at the acceptor state. We define the other pathways asmultistep 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

that flow out from L to other bridge groups and by those thatflow into L from other groups. Therefore, the amount of thecompensation, 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 limitof the single-step pathway through L. When the direction of JDLis opposite to that of JAL, the single-step contribution of L iszero. Here, we assume that the flux for the single-step pathwayis very close to the upper limit. Under this assumption, thesingle-step pathway through group L is defined numerically as

= | || | | | >

≤

‐

⎧⎨⎪

⎩⎪J

J

JJ J J J

J J

min( , ) when 0

0 when 0DL/LAsingle step

DL

DLDL LA DL LA

DL LA

(20)

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

∑≡ ℏ‐

∉

‐T JIFsingle step

L D,ADL/LAsingle step

(21)

That of the multistep pathways is defined as

≡ − ‐T T TIFmultistep

IFindirect

IFsingle step

(22)

We checked the validity of the above assumption andwhether the numerical decomposition worked correctly. Asreported in our previous article,36 TIF of the single-step andmultistep EETs exhibits polynomial and exponential decayswith increasing donor−acceptor distance, respectively. Insection 3.4, we also discuss the decay behavior of the single-step and multistep EET using the model Hamiltonian (seesection 2.4) and confirmed that the polynomial and exponentialdecays occur in the single-step and multistep EET, respectively.The least-squares method was used to fit the followingfunctions to the TIF

single‑step and TIFmultistep plots

| | = +T C R Clog log10 IF 1 10 DA 2 (23)

| | = +T C R Clog10 IF 1 DA 2 (24)

C1 and C2 are fitting parameters. The former and latterfunctions 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. Inall cases, the single-step and multistep components have a smallfitting error for eqs 23 and 24, respectively. This good fitindicates that TIF

single‑step and TIFmultistep decay polynomially and

exponentially, respectively, with the donor−acceptor distance.This result evaluated the validity of our assumption for thesingle-step pathway and demonstrated that our numericaldividing method for the single-step and multistep componentsof TIF works well.An exception is the case of #Unit = 2 in the phenylene-DB

linker systems. Because the CT states between two linker unitswere important bridge states, it was impossible for a two-unitlinker to define multistep EET. We therefore used these plotsmore than those for #Unit ≥ 3 for the fitting procedure.

2.4. Model Hamiltonian Approach for the IndirectTerms. To understand the result of the quantum chemicalcalculations for the electronic coupling and the flux analysis, we

Figure 2. The intergroup tunneling configuration fluxes of an indirectterm are numerically decomposed into single-step and multisteppathways. Arrows indicate the tunneling configuration fluxes. Thewidth of the arrow is proportional to the size of the flux.



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

vertical axes qualitatively represent a real-space coordinate andthe potential energy of local exciton states, respectively. Thehorizontal bars represent the energy levels of the excited statesinvolved in EET. Because exciton states are the primaryintermediate states in our former study,36 we only includedexciton states for the bridge states; no CT states were included.The indices D, B, and A indicate the donor, bridge, andacceptor states, respectively. Indices i and j (i, j = 1, ..., N) arefor the bridge states in the linker fragment in the order from thedonor side to the acceptor sides.The distances between the neighboring donor, linker unit,

and acceptor are a constant ΔR, which is common to all fourmodels. The distances between the donor and bridge i, bridge iand the acceptor, and bridges i and j are defined as RDi = i ×ΔR, RiA = (N + 1 − i)ΔR, and Rij = |i − j|ΔR, respectively, andare defined by parameters i and j.We assumed that interactions between these exciton states

mainly originate from the pseudo-Coulombic interaction, which

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

β β β=Δ

=−

−⎜ ⎟⎛⎝

⎞⎠

RR

iii

DD

33

(25)

β β β=Δ

= + −−

−⎜ ⎟⎛⎝

⎞⎠

RR

N i( 1 )ii

AA

33

(26)

γ γ γ=Δ

= | − |−

−⎛⎝⎜

⎞⎠⎟

R

Ri jij

ij3

3

(27)

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

∑ β β=−

⎜ ⎟⎛⎝

⎞⎠T

H 11

Eiij

jIF D A(28)

γ

γ− =

Δ ···

Δ ···

⋮ ⋮ ⋱

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

E

EH 1E12

21

(29)

ΔE is the energy gap between the donor (acceptor) and bridgestates. The potential energies of the donor and acceptor statesare the same in this model. Multiplying TIF by a constant, γ/β

2,we obtain an expression for the unitless electronic coupling TIF.

∑γβ

ββ

γ β

β = =

−⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎛⎝

⎞⎠

⎛⎝⎜⎜

⎞⎠⎟⎟T T

H 1Ei

ij

jIF 2 IF

D A

(30)

TIF has the same distance dependence as TIF; however, itdepends on only two parameters, ΔE/γ and N.

Table 2. The Least Square Fitting of the Single-Step and Multistep Components of TIF at Distances RDA to Polynomial andExponential Functionsthe Result for the Phenylene-SB Linker System

single-step multistep

polynomiala exponentialb polynomiala exponentialb

phenylene-SB slope std. dev. slope std. dev. slope std. dev. slope std. dev.

naphthalene −3.24(3) 0.0091 −0.067(4) 0.0565 −3.11(21) 0.0608 −0.0653(7) 0.0093anthracene −3.10(9) 0.0265 −0.065(3) 0.0391 −2.75(33) 0.0947 −0.059(3) 0.0372naphthacene −3.11(6) 0.0171 −0.065(4) 0.0484 −2.89(25) 0.0715 −0.061(1) 0.0181pentacene −3.08(11) 0.0313 −0.065(2) 0.0341 −2.95(31) 0.0895 −0.063(2) 0.0293

aLeast 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 andExponential Functionsthe Result for the Phenylene-DB Linker Systema

single-step multistep

polynomialb exponentialc polynomialb exponentialc

phenylene-DB slope std. dev. slope std. dev. slope std. dev. slope std. dev.

naphthalene −3.00(6) 0.0099 −0.062(2) 0.0184 −1.86(15) 0.0253 −0.039(2) 0.0133anthracene −3.04(4) 0.0064 −0.063(3) 0.0254 −2.09(15) 0.0259 −0.044(1) 0.0084naphthacene −2.99(7) 0.0127 −0.062(2) 0.0157 −2.07(25) 0.0434 −0.044(3) 0.0238pentacene −3.04(6) 0.0111 −0.063(2) 0.0184 −2.32(21) 0.0360 −0.048(2) 0.0135

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

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



To investigate the role of the interactions involved in themodel compounds, we built three models with limitedinteractions as follows. We notate the models shown in Figure3a, b, c, and d as models 3a, 3b, 3c, and 3d, respectively, forconvenience. In the first model, the interactions between bridgestates were set to zero, as in Figure 3b.

γ = 0ij (31)

Here, only the single-step coupling, which is mediated by onlyone 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 thesuperexchange theory proposed by McConnell for theintramolecular CT in the donor−(bridge)n−acceptor system.

53

γγ

== ±⎧⎨⎩

j i 1

0 othersij(32)

ββ

==⎧⎨⎩

XY D1 or NA

0 othersXY(33)

Only N-step through-bridge coupling is allowed between thedonor and acceptor, which is the opposite case from the Figure3b model. In the last model, in addition to the interactions inthe Figure 3c model, virtual exciton states are fully interactedamong the bridge states. The donor and acceptor states wereallowed to interact with the neighboring bridge states (Figure3d). This model excludes single-step coupling.Finally, the direct term is also written as eq 34 under the

assumptions above.

=Δ

= +−

−⎜ ⎟⎛⎝

⎞⎠T T

RR

T N( 1)DA 0DA

3

03

(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 Hamiltoniangiven in section 2.4, we estimated the parameter ΔE/γ. ΔE isdefined as Emonomer − Etun. The tunneling energy, Etun, is thefirst excitation energy of the donor and acceptor (Table 1).Emonomer is the excitation energy of a single bridge state that is alocal excited state of a linker unit. γ is an interaction energybetween the linker units. We examined two procedures forestimating these parameters.The first procedure is to simply use the first excitation energy

of benzene (6.31 eV) for Emonomer. In both the phenylene-SBand -DB cases, the linker unit is a benzene molecule. In thisprocedure, the Emonomer values of the phenylene-SB and -DBlinkers were treated as identical despite the differences inbonding between the linker units.We have also developed an alternative procedure, estimating

Emonomer using the Hartree−Fock state of the dimer. Using theMOD62,63 method, we transformed the dimer MOs to be veryclose to the monomer MOs. Next, a CIS calculation wasperformed 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 linkersdiffer.We estimated the interaction parameter γ between the bridge

states assuming a Frankel exciton15 model, as shown in Figure4a. The interaction γ was calculated as γ = Emonomer − Edimer or γ= ELMO‑CIS − Edimer, where Edimer is the first excitation energy of

the dimer. The γ values calculated using the former and latterprocedures were termed as γ1 and γ2, respectively, in Table 4.

For the peptide linker, we assumed that the linker unit is apeptide bond, OCNH. The first excited state of thepeptide unit is of n−π* character. The calculated interactionbetween the n−π* states of the neighbor units was so small thatthis n−π* state does not work as an effective bridge state.Alternatively, we chose the second excited state, π−π* state, asa bridge state. The π−π* excitation energy was used forELMO‑CIS, and that of the corresponding state in the peptidedimer was used for Edimer. In the actual calculations, we adoptedCH3CONHCH2CONHCH3 for the peptidedimer. Because of the asymmetry in the dimer structure,ELMO‑CIS for the left and the right CONH units weredifferent. Therefore, we averaged the left and right results forELMO‑CIS in estimating γ. In the calculation, Emonomer is smallerthan Edimer due to bonding, and γ1 is not evaluated for thislinker.The computed parameters, Edimer, Emonomer, ELMO‑CIS, γ1, and

γ2, are shown in Table 4. The calculated γ value for thephenylene-DB unit was significantly large, while that of thepeptide unit was smallest. In Figure 4b, the calculated ΔE/γvalue was compared among the donor/acceptors for each linkerunit. The ΔE/γ values for the peptide linker units were much

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

Table 4. The Excitation Energies of the Monomer andDimer in the Linker Units and Interbridge State InteractionEnergies γ1 and γ2 (See Section 2.4 for Definitions)

bridge typeEdimer(eV)

Emonomer(eV)

γ1(eV) ELMO‑CIS (eV)

γ2(eV)

phenylene-SB 5.76 6.31 0.53 6.61 0.85phenylene-DB 4.04 6.31 2.27 6.85 2.81peptide 9.89 8.87 10.05/10.09 0.18



larger than those for the phenylene-SB and -DB linker unitsbecause of the large ΔE values and the small γ value.Phenylene-SB linker systems have larger ΔE/γ than phenyl-ene-DB linker systems because of the γ value. We also find thata larger donor and acceptor have a larger ΔE/γ value becauseEtun deceases.

3. RESULTS AND DISCUSSION3.1. Direct Term Represented by a Pseudo-Coulombic

Description. In Figure 5, we compare the numerical result of

the direct term (eq 11) with that obtained with the dipoleapproximation (eq 1) for all of the molecular models. Thecenter of mass of the carbon atoms was adopted as that of thedonor and acceptor in eq 1. As shown in Figure 5, the results ofthe two calculations agree well, indicating that the direct termoriginates from the pseudo-Coulombic interaction, as assumedwhen eq 1 was derived. In other words, the exchangeinteraction 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 orientationangler factor χ (eq 2) was 2.0. These settings were chosenbecause the units in the phenylene linker align linearly and thetransition moments of the first excited states in the donor and

acceptor units are parallel to the donor−acceptor direction. Inthe models of the phenylene-DB linker, the angular factor χ is1.6−1.8, and the linearity is worse than that in the SB casebecause the linker units cannot twist. For the peptide linkers,the χ value depends on the linker length because of the softnessof the peptide.

3.2. Direct Term vs Indirect Term. The ratio of the directand indirect terms was evaluated. In Figure 6, the ratios of thecomputed direct terms in total electronic coupling TIF wereshown for all of the models. In phenylene linkers (SB and DB),the ratio of the indirect terms decreases in the order of the sizeof the donor and acceptor units (naphthalene > anthracene >naphthacene > pentacene). As shown in Table 1, a smallermolecule has a higher first excitation energy. On the otherhand, the first excitation energy of a linker unit of thephenylene, benzene, was calculated to be 6.31 eV (at the CIS/D95 level) and was larger than that of naphthalene. Thisfinding indicates that the ratio of the indirect term increaseswhen the energy gap between the donor/acceptor states andbridge 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 meansthat the ratio of the indirect term has a maximum in this region.In contrast, the ratio of the indirect term for phenylene-DBmonotonically decreased with increasing linker lengths. Thesebehaviors originate from the balance between the two distancedependencies. At longer distances, the indirect term decaysfaster than the direct term in the donor−acceptor distance.36 Atshorter cases, the number of components in the indirect termdecreases because fewer bridge states exist. This situation issimilar to the balance between the ratio of single-step andmultistep pathways, which is discussed later in section 3.4. Inthese calculations, the highest ratio of the indirect termexceeded 75%. This result indicates that the Forstermechanism14 (eq 1) significantly underestimates TIF, and theerror increases to 400% compared to that computed using aquantum chemical approach (eq 5). The total electroniccouplings TIF for phenylene-SB and -DB are shown in Figure 7aand b, respectively. The vertical and horizontal axes are in alogarithmic scale. The plots for the direct terms (dashed lines)gave straight lines, indicating that TIF shows a polynomialdecay. In contrast, the plots for the total TIF are slightly

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

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



rounded, indicating the effect of the indirect term (clearest forthe naphthalene donor/acceptor case).In the peptide linker system, the ratio changed discontinu-

ously, as shown in Figure 6c, because the orientation anglerfactor χ varies due to the donor−acceptor orientation. Thisdiscontinuity covers systematic behaviors observed in thephenylene cases, such as the distance RDA and the choice of thedonor and acceptor. In general, these systems have a greaterweight in the direct term than the phenylene linker systems. Anexceptional case is a compound with naphthacene donor/acceptor and #Unit = 5. The electronic coupling is dominatedby 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 eventhough most of the interactions between two states are drivenby the Forster type pseudo-Coulombic interaction.3.3. Single-Step vs Multistep in the Indirect Term. To

investigate components of the indirect term, we divided thecomputed indirect term into single-step and multistepcomponents using the numerical decomposition methoddescribed in section 2.3. Here, we discuss the distancedependence of the single-step ratio in the indirect term(=single-step + multistep). In Figure 8, the ratios were plotted

for the three types of linkers. Within the range of RDA, the ratioof the single-step EET was over 50%. The result clearly showsthat single-step EET is the major component of the indirectterms in the bridge mediated EET.In phenylene-SB with naphthalene as the donor and

acceptor, the indirect term was the major component. Within2 ≤ #Units ≤ 6, the indirect term comprised more than 60% ofthe total TIF, as shown in Figure 6a. Over 42% of TIF of thecompound originates from the single-step terms, less than 40%from the direct term, and the rest (approximately 20%) fromthe multistep terms. The importance of the multistep EETincreases in the phenylene-DB case. For example, in the case of#Units = 6 and naphthalene as the donor and acceptor, thedirect term was just 25% of the total TIF, and approximately42% was from the single-step EET. The remaining 33% arosefrom the multistep EET. These particular cases show that thesingle-step EET is the major component even in singlet EETand 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 Figure8c. However, because the direct term was very dominant in thetotal TIF, the single-step term was a minor contributor in thisseries of compounds. The only exception was the case for#Units = 5 with a naphthacene donor/acceptor. The calculateddirect term was small because of the orientation factor χ in thecomputational model. In this case, the single-step term shared61% of the total TIF, while the direct term shared 30%. We notethat the effects from the donor−acceptor orientation did notclearly appear in the indirect terms, which is probably becausethe summation of many indirect terms averaged the variation inthe 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 thephenylene-SB result with the DB one because we are interestedin how the interactions between the fragments affect themechanism. If a fragment is divided into small pieces offragments, 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 thesingle-step ones increases. However, such multistep pathwaysrun only the local region of the bridge, and a “single-step”-likeview is unchanged. In addition, the present definition of the

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

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).



fragment is reasonable because, in the phenylene-DB, themultistep 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 indirectterms in Figure 6. In particular, the behaviors of the plots inFigure 8a and b are similar to those in Figure 6a and b,respectively. For the phenylene-SB case, the curves are convexdownward. For the phenylene-DB case, the curves decreasemonotonically. These behaviors depend on the multistepcontribution, which will be analyzed using the modelHamiltonian in the next section.3.4. A Model Hamiltonian Study for the Indirect EET.

The quantum chemical calculations for the model compoundsconsidered all possible bridge states and interactions within agiven theoretical model. However, because of this generalityand complexity, the result is not easily interpreted in terms ofphysical parameters such as energy differences and interactionsbetween the units. To obtain a clear view of the indirect EET,we introduced a model Hamiltonian including simple physicalparameters as described in section 2.4. Because TIF dependsonly on two parameters, ΔE/γ and N (the number of bridgeunits), we can explore these parameters in the modelcalculations. We are particularly interested in how the distancedependence of TIF (eq 30, the distance-dependent factor ofTIF) behaves under the variation of the EET models (from 3ato 3d in Figure 3) and the parameters (ΔE/γ and N). Thisinformation could also help us understand the results forvarious donor/acceptors and linker units.First, we used the “fully interacted” model 3a (Figure 3a) to

investigate how the TIF value decays with the number of thelinker units (from N = 0 to 100). The results of the calculationswith three different ΔE/γ values (1.85, 2.00, and 5.00) areshown in Figure 9a. For a larger ΔE/γ value, such as ΔE/γ =5.0, the TIF decay line is straight. In contrast, when ΔE/γdecreases, the decay line became rounded in the region from N= 3 to 50 (log10N = 0.5−1.7). The superexchange mechanismfrequently breaks down for much smaller ΔE/γ values causedby the resonance between the bridge energy and the tunnelingenergy.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 EETwas limited to single-step EET. As shown in Figure 9b, model3b produced a decay “line”, indicating that the single-step EETgives a polynomial decay. From the slope of the decay line, theTIF value is proportional to N

−3. This result is explained by eachstate-to-state interaction being proportional to N−3 in thepresent model. Model 3b is a limit of the weak interunitinteraction. In model 3c (McConnell type superexchangeregime), only nearest neighbor interactions were allowed in theEET. Thus, only (N + 1)-step (multistep) EET was involved.The result in Figure 9b shows a rounded decay, which is verysimilar to the rounded decay seen in model 3a with smallerΔE/γ values. This exponential decay is characteristic ofmultistep EET, as we described in a previous study.36

Therefore, the rounded decay curves in model 3a originatefrom the multistep EET. In model 3d, we only allowed thedonor and acceptor to interact with their neighbor linker unit;no single-step EET was involved. As seen in Figure 9b, thedecay curve for shorter distances was dominated by theexponential type characteristic of the multistep EET, and thecurve is very similar to that of the full model 3a. The multistepexponential decay is caused by the increase in the number of

the steps, Nstep. On the other hand, the decay of the multistepcoupling also includes a polynomial decay. This polynomialdecay is because the distance between the donor and acceptorincreases under the same Nstep. Such a fixed-Nstep effect resultedin a polynomial decay in the TIF 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 thetotal TIF. In the popular McConnell type superexchange53

model designed for CT, Nstep always is equal to N + 1 andincreases with the distance. The TIF value, therefore, decaysexponentially (see model 3c in Figure 9b). However, in EET,the couplings with Nstep ≠ (N + 1) dominate and the TIF decaysare polynomially caused by the fixed-Nstep effect.Considering the behaviors of models 3b−3d, the N

dependence of TIF in model 3a (the full model) arises fromthe superposition of the exponential and polynomial decays.The former originates from multistep pathways that areenhanced when the number of steps increases. The multistepEET was the predominant portion of the indirect term forintermediate N values. The latter comes mainly from the single-step pathways at increasing RDi and RiA distances. Because theexponential terms decay faster than the polynomial terms in theregion of large N, the rounded decay curves shown in Figure 8arevert to the straight form at a certain N value. We note that in

Figure 9. The distance factor TIF (eq 25) for indirect EET versus thenumber of the bridge units (N). The donor−acceptor distance isdefined as (N + 1)ΔR. (a) ΔE/γ dependence. (b) Model dependenceat ΔE/γ = 2.0.



the limit of small N, the number of bridge states is insufficientto 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 theresult for the phenylene-SB linker, both the single-/multistepratio and the direct/total ratio decreased at intermediate linkerlengths because the multistep components have a significantcontribution in that region. In the results of the phenylene-DBlinker, both the single-/multistep ratio and the direct/total ratiodecayed monotonically with increasing linker length. Becausethe ΔE/γ value of the phenylene-DB linker is smaller than thatof phenylene-SB, the multistep region extends to a greaterlinker length region and the ratio of the multistep EETdecreases 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 tunnelingconfiguration flux. Here, we return to the original Hamiltonianand calculate the intergroup tunneling configuration flux for thephenylene linker (SB and DB) systems (#Units = 3) with ananthracene donor/acceptor. For convenience, the three linkerunits shown in Figure 10b and c were labeled as i, ii, and iiifrom left to right in the figure. We first show that the majorbridge states are exciton groups (intraunit excited states) andCT groups (interunit CT excited states) between neighboringunits. Other CT states were neglected from the flux for thepresent purpose. These five exciton and eight CT groups werelabeled 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 approximateposition of the excitations in the molecule. We next defined acoordinate z. These 12 excitation groups were aligned along thez axis in the above order, as shown in Figure 10a. The z valuewas incremented between neighboring two groups. The donorgroup 1 and acceptor group 7 are located at the left and right

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

∑ ≡∈∈

K z K( )z

zL left ofM right of

excitonneighboring CT

LM

(35)

At a specific z value in the one-dimensionally aligned groups,K(z) is defined as the summation of the tunnelingconfiguration flux from a group in the left-hand side of z to agroup in the right-hand side of z. K(z) represents a tunnelingconfiguration flux that passes a border defined by z.The computed K(z) is shown in Figure 10a. Although only

exciton and neighboring CT groups were considered, the K(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 suggestedthat the exciton and neighboring CT groups are importantbridge states. In addition, the direct term contributes only 49and 40% in the SB and DB cases, respectively, as shown inFigure 6. Within the indirect term, the single-step pathway wasimportant, as shown in Figure 8, and we can conclude that thedominant pathways in the bridge-mediated EET are direct andsingle-step EETs via the exciton states and the neighboring CTstates.The main intergroup tunneling configuration fluxes with |

KLM| > 0.01 for the phenylene linkers (SB and DB) are drawnusing arrows in Figure 10b and c, respectively. The width of thearrow is proportional to the amount of the tunnelingconfiguration flux. The direct flux (blue arrow) is predominantin all fluxes. The red arrows are through-exciton fluxes andinclude all pure through-exciton pathways. Exciton states areshown as the “ex” index in the figures. The orange arrows pass

Figure 10.Major EET tunneling pathways in the phenylene linker models. (a) The K(z) plot. See text. Major tunneling fluxes for the (b) phenylene-SB and (c) phenylene-DB linkers (|KLM| > 0.01). Blue, red, and orange arrows denote the direct, through-exciton, and through-neighboring-CTfluxes, respectively. Green arrows are through-CT (from linker units i to iii) fluxes. Exciton states are shown using index “ex” with red circles. Inparticular, 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 thearrow is proportional to the magnitude of KLM.



through the neighboring CT groups. CT states are shown usingthe “+−” index in the figures, and relative positions of + and −indicate the direction of CT separation. We see the relativelylarge single-step pathway fluxes: In both SB and DB, the arrowsstart from D, pass one of the linker units, and finish at D. Thephenylene-SB linker has no other dominant pathway, as shownin Figure 10b. In contrast, for the phenylene-DB linker, a CTstate between distant units (see green arrows in Figure 10c)also contributed, which represents the difference of theinterunit 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 tunnelingpathways (configuration fluxes) are related to the phases anddetermine the interference among the tunneling pathways. Thiswas discussed for the electron transfer.60,72−76 The tunnelingpathways with an opposite direction destructively interfere anddecrease the total TIF. In contrast, one with the same directionconstructively interferes and increases the total TIF. In Figure10b and c, all of the fluxes run from the donor to acceptor sides,and these tunneling pathways enhance the total TIF. In ourprevious studies,36,38 destructive interference was observed inan EET mechanism.

4. CONCLUSIONS

In general, singlet EET has been explained by a Forstermechanism that includes a direct electronic coupling betweenthe donor (D) and acceptor (A) states. Considering the long-range EET observed in biology and nanotechnology, thepossibility of bridge-mediated indirect EET, so-called super-exchange EET, is of particular interest and is well suited fortheoretical studies. In this study, we have explored conditions inwhich the superexchange term in EET dominates andsystematically examined various donor/acceptor, linker types,and linker lengths in the donor−linker−acceptor molecularmodels using the quantum chemical approach developed in ourprevious study.36

The indirect term in the electronic coupling becomesmaximal at a particular distance caused by the following twoeffects. Although the number of the indirect terms increaseswith the distance, the indirect terms decay more rapidly thanthe direct terms at longer distances. In the phenylene-DB case,the indirect term became up to 4 times larger than the directterm. We numerically decomposed the indirect term into thesingle-step and multistep components. In either very small orlarge D−A distances, the single-step term was more dominantthan the multistep term. The multistep term became dominantonly in intermediate distances because of the exponential decaybehavior.We also developed a one-dimensional simple model for the

bridge-mediated singlet EET for comparison with the results ofthe quantum chemical models. When the potential energies ofthe local bridge states are close to the tunneling energy and theinteractions between the bridge states are strong enough, theindirect ratio exceeds the direct ratio. The qualitative behaviorobtained by quantum chemical calculations was reproduced bythe simple model Hamiltonian, in which we assumed only localexciton-like bridge states and pseudo-Coulombic interactionsamong donor, acceptor, and bridge states. Although the nearestneighbor interaction model is known as McConnell typesuperexchange for the CT, single-step interactions areimportant for the singlet EET. Therefore, the TIF for singletEET of the superexchange type decays dominantly in a

polynomial way, which is different from the exponential decayseen in the McConnell type superexchange model.The EET pathways were also analyzed. In phenylene linker

systems, bridge-mediated EET pathways are single-step mostlyeither through-exciton or through-neighboring CT states. Theexciton states (local excited states in the linker units) aredominant in EET, especially when the interactions between thebridge states are small.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected]. Phone: +81-75-711-7867. Fax: +81-75-781-4757.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis study was supported by KAKENHI (No. 21685002) fromthe Japan Society for the Promotion of Science (JSPS), JST-CREST, and Strategic Programs for Innovative Research(SPIRE). Some of the computations were performed atRCCS (Okazaki, Japan) and ACCMS (Kyoto University).

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Singlet Excitation Energy Transfer Mediated by Local Exciton Bridges