Bridge-Mediated Excitation Energy Transfer Pathways through Protein

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Bridge-Mediated Excitation Energy Transfer Pathways through Protein Media: a Slater Determinant-Based Electronic Coupling Calculation Combined with Localized Molecular Orbitals Tsutomu Kawatsu,† Kenji Matsuda,† and Jun-ya Hasegawa*,†,‡ †

Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Kyoto-Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan ‡ Quantum Chemistry Research Institute (QCRI) and JST-CREST, Kyodai Katsura Venture Plaza,1-36 Goryou Oohara, Nishikyo-ku, Kyoto 615-8245, Japan

bS Supporting Information ABSTRACT: A computational method for calculating electronic coupling and pathway of electron transfer (ET) has been extended to that for excitation energy transfer (EET). A molecular orbital (MO)-based description has been generalized to one based on Slater determinants. This approach reduces the approximations used for the Green’s function method from the perturbation of chemical-bond interactions to the perturbation of the configuration interactions. It is, therefore, reasonable to apply this method to EET, which involves the transfer of an electronhole pair. To represent EET donor, acceptor, and bridge states, we adopted recently developed localized molecular orbitals (LMOs) for constructing locally excited determinants. The LMO approach provides a chemically meaningful interpretation of how each local excitation on the bridge contributes to the total electronic coupling of the EET. We applied the method to six model peptides and calculated their electronic couplings as well as the direct and through-peptide terms. Although the through-peptide term is usually negligibly small compared with the direct term, it can dominate the EET reaction in appropriate situations. The direct term dominates in long-range interactions because the indirect term decays in shorter distances.

1. INTRODUCTION Excitation energy transfer (EET) is a fundamental process in photochemistry and photobiology. In photosynthesis, excitation energy is transferred from an antenna system to the photosynthetic reaction center.15 EET is also a key process in industrial products such as light emitting diodes (LED),6,7 solar cells,8,9 and lasers.10 In fluorescence (or F€orster) resonance energy transfer (FRET),11 which has become an important analytical tool in molecular biology,12 EET quenches an emission of an excited molecule and provides an alternative decay pathway such as fluorescence or phosphorescence. Various EET theories have been proposed and applied to EET phenomena.1316 Particularly, the EET in the photosynthetic system has been extensively studied.1730 Both EET and electron transfer (ET) depend crucially on the electronic coupling between the initial and final states. In the weak coupling limit, the magnitude of the coupling determines the reaction rate.31 The electronic coupling for EET includes Coulombic and exchange terms. In particular, EET processes driven by longrange Coulombic interactions and short-range electron exchange interactions are referred to as F€orster transfer11 and Dexter mechanism,32 respectively. In the strong coupling regime, in contrast, the initial and final states are resonant, and decoherence r 2011 American Chemical Society

processes determine the reaction rate.33,34 Systems with the intermediate coupling regime have also been studied.24 In a weak coupling regime such as long-range EET, the electronic coupling between two excited states is approximated by the first order terms in the multipole expansion and is expressed with the dipoledipole interaction as11 dipole

T ab

¼

χ jma jjmb j R3

ð1Þ

where ma and mb are the transition dipole moments of states a and b, respectively, and R is the center-to-center distance between the donor and acceptor. The angular parameter χ is written as χ ¼ cos θab 3 cos θaR cos θbR

ð2Þ

where θab, θaR, and θbR are angles between ma and mb, ma and the direction of EET, and mb and the direction of EET, respectively. The decay of the dipole interaction with R3 imparts its longrange character. On the other hand, short-range electronic Received: July 19, 2011 Revised: August 23, 2011 Published: August 23, 2011 10814

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couplings, such as those in charge transfer interactions, decay exponentially with R. Electronic coupling is generally influenced by atoms and molecules between the donor and the acceptor3541 (hereafter, we call such a mediator as a bridge). Bridge-mediated EET has been studied in various linked chromophores both theoretically and experimentally.4252 Through-bond interactions are often important in the triplettriplet EET5052 because of the shortrange of the Dexter mechanism and sometimes dominate singletsinglet EET.4349 Evidence of charge transfer (CT) state-mediated EET has been observed experimentally.44 In previous theoretical studies, a photon53 and CT states between the donor and acceptor sites54 were included as the bridge configuration. However, when there is a bridging molecule, the excited and CT states involving the bridging molecules should be considered. To the authors’ knowledge, general excited configurations have never been included as the bridge states in bridgeddonor/acceptor EET models. When bridge configurations are energetically separated from the donor and acceptor excited states (more than the energy of thermal fluctuations), EET occurs via the superexchange mechanism; otherwise, the excitation energy can be transferred through real intermediate states. Transfers through the intermediate states have been studied for molecular aggregates such as light-harvesting systems.2630 In the present paper, we report on EET in the superexchange regime. We used the second order perturbation method41,5559 for calculating the bridge-mediated electronic coupling (see section A in Supporting Information for details). ^ Fæ T IF ðEÞ ¼ ÆΨI jE HjΨ

density tunneling flux,” or simply “tunneling flux.” The tunneling flux from configuration p to q is given by63 (see section B in Supporting Information for details) 1 J pq ¼ ðC̅ Ip C̅ Fq C̅ Fp C̅ Iq ÞH pq p

where C is a CI coefficient, and superscripts I and F denote the initial and final states, respectively. Determinants used here are orthogonal each other. This formalism allows investigation of the roles of the bridge configurations with established techniques of the tunneling current analysis. In the next section, we describe theoretical and computational details of bridge-mediated electronic coupling calculations and the tunneling flux analysis for EET. Computational results and the discussion are presented in section 3, followed by the conclusion in section 4.

2. THEORY AND COMPUTATIONS 2.1. Localized Molecular Orbital (LMO) Approach for Electronic Coupling and Tunneling Flux. Slater determinants

were constructed from LMOs obtained with a hybrid60 of the minimum orbital-deformation (MOD)69 and the population localization methods.70 This approach allows definition of initial and final states that are localized on the donor and acceptor molecules, respectively. This description also lends itself to a chemically motivated interpretation of the pathway determined by interconfiguration tunneling flux. The LMO is written as a linear combination of atomic orbitals (AO) as ¼ ϕLMO r

ð3Þ

ΨI and ΨF are the wave functions of the initial and final states, ^ is the electronic Hamiltonian, and E is the respectively. H transferred energy. In our previous studies on ET,59 one-electron molecular orbitals (MOs) were used for ΨI and ΨF. In this study, we generalized the wave functions: singly excited determinants were used for studying EET. We also adopted localized molecular orbitals (LMOs).60 In the localization scheme, we use a predefined set of reference MOs (RMOs). Delocalized canonical MOs (CMOs) are transformed to be very similar to the RMOs.60 These LMOs clearly define the initial, final, and bridge states, including localized exciton and charge transfer between certain local sites. Here, we briefly compare the present calculation with those of previous studies. Hayashi and Kato61 and Nishioka and Ando62 introduced Slater determinants for studying bridge-mediated electronic coupling for ET through valence orbitals. To analyze the ET pathway, LMOs were also adopted.61,62 Regarding EET, Harcourt and coworkers introduced only CT configurations between donor and acceptor,54 but no other bridge configuration was considered. Estimating the bridge’s effects on the electronic coupling is also important for analyzing EET processes. Tunneling current analysis is useful for describing the virtual pathways involved in superexchange electronic coupling through media.6368 Although such analyses have been applied to ET reactions,6368 they have not been applied to EET process because the tunneling current is defined as an electronic flux from the donor to acceptor molecules. Here, we have extended the concept of a tunneling current from an electronic density flux to a density flux of electronic configurations, referred to here as “configuration

ð4Þ

LMO ∑ν ϕAO ν dνr

ð5Þ

The population of the LMO on each fragment is computed using L€owdin orbital population analysis71 and is given by Fnr ¼

LMO LMO 1=2 S1=2 ∑ ∑ ημ dμr d νr Sνη η ∈ n μν

ð6Þ

for LMO r on fragment n. LMO r is defined as “belonging” to a single fragment n when the population Fnr is the largest of all fragments. For convenience, we define a “belonging” function, ( 1 if r belongs to n n ð7Þ ϑr 0 else We construct the singly excited determinant using LMOs as the one-electron basis. A series of determinants can be written with a pair of the electron creation and annihilation operators, LMO† LMO âi Φ0

¼ â a ΦLMO ai ¼ ¼

fragment LMO

ϑna ϑm aLMO âLMO Φ0 ∑ ∑ a i ^ i ai nm

fragment LMO

∑ ∑ai ΦLMO ai=nm nm

†

ð8Þ

where Φ0 is the Hartree-Fock (HF) ground-state determinant, and indices a and i denote occupied and virtual orbitals instead of the determinant index p in the previous section. We introduce an expression of the determinant ΦLMO ai/nm, where an electron from occupied orbital i in fragment m has been excited to virtual orbital 10815

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a in fragment n: †

n m LMO LMO ΦLMO aa âi Φ0 ai=nm ϑa ϑi ^

ð9Þ

The initial and final states ΨX (X = I, F) are written as ΨX ¼

∑ ∑ CXai ΦLMO ai=nm ∑ ΨX=nm nm ai nm

ð10Þ

The values of the CXai were computed using Green’s functions with a second order perturbation of the configuration interaction between the major donor/acceptor and bridge determinants (see section A in Supporting Information). Excitations involving acceptor and donor were excluded in calculations of the initial (X = I) and final (X = F) states, respectively. In the LMO basis, the electronic coupling element is given by ^ F=lk æ ∑ ÆΨI=nm jE HjΨ ^ LMO ̅ Fbj ¼ ∑ ∑ C̅ Iai ÆΦLMO ai=nm jE HjΦbj=lk æC nmlk ai, bj

T IF ðEÞ ¼

nmlk

lk , ∑ T ai,IF bj nmlk ∑ T nm, IF ai, bj

ð11Þ

where nm and lk define a set of one-electron transitions from fragment n to m and from fragment l to k, respectively. The electronic coupling is, therefore, decomposed into contributions at the fragment level. Using this notation, the tunneling flux is given by J ai, bj

1 I F ðC̅ C̅ C̅ Fai C̅ Ibj ÞH ai, bj p ai bj 1 I F ^ LMO ðC̅ ai C̅ bj C̅ Fai C̅ Ibj ÞÆΦLMO ai=nm jHjΦbj=lk æ p nmlk

¼

∑ lk J nm, ∑ ai, bj , nmlk

¼

ð12Þ

and the total tunneling flux from the transition nm to lk is J nm, lk ¼

lk J nm, ∑ ai, bj ai, bj

ð13Þ

Note that Jnm,lk = (Tnm,lk Tlk,nm IF IF )/p and Jnm,lk = Jlk,nm. These fluxes are conservative, except at the source (donor) and sink (acceptor). We employ a dimensionless normalized tunneling flux K nm, lk p

J nm, lk T IF

where the conservation rule becomes 8 > < 1 ðY ¼ DÞ K nm, lk ¼ 1 ðY ¼ AÞ > nm ∈ Y lk ∈ all : 0 ðothersÞ

∑ ∑

ð14Þ

ð15Þ

The amount of the tunneling flux entering the fragment Y is the same as that running out of the fragment Y, except for the Y = D and Y = A cases. This normalized tunneling flux is used in following discussions. We also note that the present theoretical framework is applicable to analyze triplettriplet energy transfer, although some modifications are necessary in matrix element evaluations. 2.2. Model Systems. We calculated the electronic coupling of the model peptide systems shown in Figure 1. The models were constructed from four aromatic amino acids (two phenylalanine

Figure 1. Structure of a model peptide, Model 14, with tyrosines located at fragments 1 and 4 (C7H7O-CH2-CO-Phe-Phe-NH-CH2C7H7O). Fragments for the electronic coupling calculation and tunneling flux analysis are divided by white dotted lines. The picture was rendered with GaussView4.1.72.

and two tyrosine residues). Both peptide terminals were truncated at the next α carbon and capped by a hydrogen atom. We prepared six models with varying phenylalanine and tyrosine positions. Model nm denotes the model peptide with tyrosine residues for fragments n and m. Each model molecule was divided into seven fragments as shown in Figure 1. Fragments 1 to 4 include the side chain and α carbon of an amino acid, and fragments 5 to 7 include a peptide bond. We calculated electronic couplings between the first excited states of two tyrosine fragments. Because tyrosine’s transition energy is smaller than that of phenylalanine, the electronic coupling was expected to fall in the superexchange regime. In our preliminary calculations, the first excitation energy of the phenol moiety was 6.17 eV, which is lower than the first excited states of benzene (6.30 eV) and the peptide (6.39 eV) at the CIS level. 2.3. Computational Details. A (Phe)4 molecule was prepared as a structural template. The structure was built using GaussView4.172 and optimized at the B3LYP73,74/6-31G(d)75 level using Gaussian03.76 After optimization, a hydrogen atom in phenylalanine was replaced by a hydroxyl group to create tyrosine. Following this replacement, only the H and OH parts were reoptimized to avoid introducing other complexities (such as effects from orientation and orbital distributions) into the present analysis. Calculations of the electronic coupling and tunneling flux were performed with singly excited determinants using the HF MOs. The D95 basis sets77 were employed. These determinants were based on LMOs localized on each fragment in the model peptides (see Figure 1).60 For fragments 1 to 4, MOs were localized using the MOD method,69 which transforms CMOs to LMOs that strongly overlap a given set of RMOs. RMOs were calculated for a single fragment truncated from the model peptides with the D95 basis set and capped by a hydrogen atom with the STO-6G78 basis set. After the MOD process, the remaining orbitals surrounding the peptide bonds (fragments 57) were also localized using the population localization method.70 As discussed below, the LMOs produced by the present scheme are localized on each predefined fragment. Because we did not mix occupied or unoccupied spaces, the excitation space spanned by the CIS determinant was identical to those in the conventional CIS wave function. For major donor and acceptor determinants in the electronic coupling calculations (D0 and A0 in section A in Supporting 10816



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3. RESULT AND DISCUSSIONS 3.1. Distance Dependence of TIF. We computed the electronic couplings and tunneling fluxes for the six model systems as described in section 2.2. All models had tyrosine as both the donor and the acceptor species with various relative distances and orientations. Distances were computed as the distance between the centers of the six-membered rings in the donor and acceptor tyrosines. The electronic coupling TIF for EET is driven by Coulombic (F€ orster mechanism11) and exchange (Dexter mechanism32) interactions. The Coulombic interaction is long-range and proportional to R3 (in the multipole expansion), where R is the distance between the donor and the acceptor. On the other hand, the exchange interaction is short-range. Both of these interactions are included in the Hamiltonian. TIF can be divided into direct and indirect interactions,

T IF ðEÞ ¼

lk lk T nm, þ ∑ T nm, ∑ IF IF nm ∈ D, lk ∈ A others

ð16Þ

The first and second terms on the right-hand side are the direct and indirect interactions, respectively. It is noteworthy that bridge-mediated terms in the electronic coupling will decay faster than the direct term ( R3) when the sitesite interaction decays with R 3 (see section C-1 in Supporting Information).

Figure 2. Computed electronic coupling TIF vs donoracceptor distance. E = 0.2216 au was used. Black triangles, red circles, and green diamonds represent the total, direct, and indirect TIF terms, respectively. The dotted (χ = 0.25) and broken magenta (χ = 0.005) lines represent the dipole model with fixed χ values. The green solid line is the leastsquares fit for the indirect terms. The donoracceptor distance R is calculated as the distance between the centers of the six-membered rings.

The computed total TIF values for the six models, with the corresponding donoracceptor distances R, are shown in Figure 2 and are decomposed into direct and indirect terms. Except in models 23 and 34, the direct terms dominated, with the indirect term being almost negligible (representing only a few hundredths of the direct term). In model 34, the direct term was very small, considering the donoracceptor distance (R = 8.5 Å). This attenuated direct term is due to a small χ value arising from the orientations of the six-membered rings in the donor and acceptor. In this case, the indirect term was approximately 44% of the direct term and opposite in sign, leading to cancellation in the total TIF value. The indirect term of model 23 was 30% of the direct term. Figure 2 also shows two lines representing electronic = (χ/R3) couplings calculated using the dipole model Tdipole DA |mD mA| with fixed χ values. |mD| and |mA| were also fixed to 1.4 e 3 au. The dotted line indicates the large χ case (χ = 0.25), which qualitatively impacts the total value of TIF. The magenta broken line represents the small χ case (χ = 0.005). In model 34, the unfavorable orientation of the donor and acceptor fragments reduces the total TIF value. The direct term becomes competitive to the indirect term, and the indirect term could affect the EET reaction. A least-squares fitting of the indirect term is shown as a green solid line in Figure 2, giving a dependence proportional to R4.1. This exponent is smaller than that of the dipole model (3.0) (see section C-1 in the Supporting Information). Therefore, a much smaller χ value is required for bridge-mediated couplingdominated EET at longer distances. 3.2. Direct Term in the Electronic Coupling Compared with the Dipole Model. The direct donor/acceptor coupling term is compared to that of the dipole model, Tdipole DA . Only the direct terms between major donor and acceptor configurations (D0 and A0) were considered. The electronic coupling in the dipole model was computed for the HOMO to LUMO transitions in the donor and acceptor fragments. The mD and mA are computed using HOMO and LUMO, which were calculated as the RMOs of corresponding fragment. The norms of transition dipole moments, |mD| and |mA|, were 1.4 e 3 au. The CI )

Information), we chose determinants describing electronic transitions from the highest occupied MO (HOMO) to the lowest unoccupied MO (LUMO) in the tyrosine fragments. These determinants were dominant in the CIS wave functions of the first excited states of each fragment. All other determinants locally excited within the tyrosine fragments were defined as minor donor and acceptor determinants for EET (D and A in section A in Supporting Information). Because all LMOs belonged to single fragments, singly excited determinants were categorized by a pair of fragments. These determinants can be classified as excitonic when the pair is on the same fragment, or as charge transfer (CT) if they fall on different fragments. E = 0.2216 au, the average energy of the first excited states for the four tyrosine fragments in CIS level, was applied as the transferring excitation energy. In the perturbed calculations, coefficients for the major donor and acceptor determinants ranged from 0.964 to 0.968. The secondary component of the first excited state was the HOMO1 to LUMO+1 transition determinant, and the absolute coefficient value is approximately 0.23. In the original CIS calculations for each tyrosine fragment, the absolute value of the coefficient for the HOMO to LUMO determinant was approximately 0.87. The rest of the population fell mostly into the HOMO1 to LUMO+1 determinant, with an absolute coefficient value of more than 0.47. Differences in the coefficients calculated by the two methods are due to the absence of the higher order perturbation terms in the configuration interactions (see section A in Supporting Information). However, these differences do not affect the present conclusions. Electronic couplings were also calculated using the dipole dipole interaction model (hereafter, the “dipole model”) for comparison. For these calculations, the transition dipole moments of the excited states were calculated using the RMOs described above.

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Figure 3. Direct term of the computed electronic coupling compared with that obtained from the dipole model.

Table 1. Angular Parameters χ for Pairs of Donor/Acceptor Configurations χ (D0/A0)a

χ (D0/A1)a

χ (D1/A0)a

χ (D1/A1)a

Model 12 Model 13

0.55 0.81

0.54 0.82

0.54 0.81

0.54 0.82

Model 14

0.66

0.66

0.66

0.65

Model 23

0.28

0.25

0.29

0.27

Model 24

1.07

1.09

1.09

1.11

Model 34

0.04

0.03

0.06

0.05

a D0 and A0 are HOMO to LUMO excitation configurations, and D1 and A1 are HOMO1 to LUMO+1 excitation configurations.

coefficients CIDH DL CFAHAL = (0.97)2 were multiplied with the results of the dipole model to allow comparison with the present ab initio model. Results are shown on the horizontal and vertical axes, respectively, of Figure 3. There was good agreement between the two models, despite the disparate χ values among the six models because of the differing tyrosine ring orientations (see Table 1). These results indicate that the direct coupling term is mainly driven by dipoledipole interactions. Note also that the exchange interaction is not important for long-distance electronic couplings, such as those that occur in these model peptide systems, because the dipole model does not include exchange interaction. Table 1 shows the angular parameter χ for the major donor and acceptor (D0 and A0) and secondary donor and acceptor (D1 and A1) for all models. Because the transition dipole moments of D0 and D1 (and A0 and A1) are almost parallel to each other, the absolute values of χ were similar but with the opposite sign. Model 34 had a much smaller value of χ than the other models, explaining why it has the smallest direct term in Figure 2. 3.3. Intermediate States in the Indirect Terms. We computed the normalized tunneling flux for all model systems to investigate the indirect EET coupling pathway. The system was separated into seven fragments, and the singly excited determinants were classified into 49 (=7 7) types. These determinants included excitonic and interfragment CT configurations, as shown in the list of Figure 4d. Figures 4a, b, and

c show the large computed tunneling fluxes for models 12, 14, and 34, respectively. These cases represent short distance (model 12), long distance (model 14), and small direct term (model 34) couplings. In Figure 4, we introduce a superindex notation N = nm and M = lk for the normalized tunneling flux from a coarse-grained point of view, KNM = Knm,lk. For example, groups N = 1, 9, 17, 25, 33, 41, and 49 were exciton states within fragments 1, 2, ..., 7, respectively. Group N = 45 was a CT state from fragments 7 to 3. The vertical and horizontal axes of Figures 4a, b, and c indicate these superindices. The color scale from red to orange shows logarithms of positive values of KNM, while the color scale from blue to cyan indicates the logarithms of the absolute values of negative values of KNM. These values appear antisymmetrically in the matrix because KNM = KMN. The columns and rows of the matrix are the input and output of the tunneling flux for each group. As shown in Figures 4a, b, and c, we obtained a very strong direct flux from the donor to acceptor, and the indirect fluxes were relatively small. However, the behavior of the coupling pathway is worth investigating because the direct term can be small in situations such as model 34. Model 12 is a short distance coupling model. In this model, the distance between the centers of fragments 1 and 2 was 10.6 Å. The donor and acceptor were groups 1 and 9, which represent exciton states on fragments 1 and 2, respectively. Tunneling flux for the direct term, K1,9, is shown as “1 f 9” in Figure 4a, which is the largest component in the total flux. The most dominant indirect tunneling fluxes were through groups 33 and 41 (see Figure 4a) with two steps. For example, there was a sequential flux from the donor 1 to the group 41 (1f41) followed by the flux from the group 41 to the acceptor 9 (41f9). This two step flux is the same direction to the direct term and increases the amount of the total flux. On the other hand, there was a flux from the acceptor 9 to donor 1 (cf. a flux 9f33 followed by 33f1). This flux is an inverse direction and cancels a part of the direct term. We also found a three step flux from the acceptor 9 to the donor 1, 9f29 to 29f33 to 33f1, which also partially cancels to the direct term. Group 33 was an exciton state at the peptide bond in fragment 5, which is located between the donor and acceptor. Group 41 was also an exciton at the peptide bond in fragment 6, which is adjacent to the acceptor. There are also contributions from groups 17 and 49, exciton states on fragments 3 and 7, respectively. There was little contribution from group 25, possibly because that group is an exciton at fragment 4, which is far from both the donor and acceptor. Groups 5, 12, 13, 29, 30 and 37 also mediated the electronic coupling. These were CT states involving the donor, acceptor, and their neighboring peptide bonds. The electronic coupling involved in CT is short-range, and these types of tunneling fluxes appear in the short distance models. In short distance cases such as model 12, CT states function as bridge states. When a CT state involves fragments between the donor and the acceptor, this term represents the Dexter mechanism. As seen in Figure 4a, a CT state, group 5, is from the donor fragment 1 to fragment 5. The transferred flux was then transferred to group 9, the final state of EET. Figure 4b shows results for model 14, a long distance case. The donor and acceptor were groups 1 and 25, respectively. The center-to-center donor/acceptor distance was 17.6 Å. In the indirect terms, exciton-mediated two-step tunneling fluxes dominated, as shown in Figure 4b. All exciton groups (9, 17, 33, 41, 10818



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Figure 4. Normalized tunneling flux, KNM, having large absolute values. The donors and acceptors in each system are the local excited states (excitons) defined in (d). The red to orange color scale shows the log of KNM, and the blue to cyan color scale represents its antisymmetric components. E = 0.2216 au was used for all calculations. (a) Model 12, a short-distance case. The donor and acceptor were groups 1 and 9, respectively. (b) Model 14, a longdistance case. The donor and acceptor were groups 1 and 25. (c) Model 34, a short-distance case with a small direct term. The donor and acceptor were group 17 and 25. (d) List of groups (N and M). The notation nfm denotes a single-electron transition from an occupied MO in fragment n to a virtual MO in fragment m.

and 49) mediated tunneling fluxes. This is because the dipole dipole interactions between the exciton states survived in the long-range interaction as described in subsection 3.2. The tunneling flux through group 17 was weak. This is because the ring orientation between fragments 3 and 4 made a part of the transition dipole interactions weak (see Table 1) and also because the benzene ring of fragment 3 was not close to the straight line connecting the donor and acceptor, limiting its usefulness as a pathway. For model 34, the direct term was small because of the orientation between the aromatic rings of the donor and acceptor fragments. Model 34, like model 12, is a short distance case. The distance between the donor and acceptor was 8.5 Å. The computed direct term for model 34 (1.7 105 au) was small compared to other models (for example, 3.6 104 au for model 24, which had a similar donor/acceptor distance). Because the direct term was small, many indirect KNM terms became relatively important. Even a CT state between the donor and the

neighboring peptide, group 45 (7f3), contributed to the electronic coupling as much as group 49 (7f7), an exciton state between the donor and acceptor. As discussed above, some CT states indicate a Dexter mechanism. The tunneling fluxes for the Dexter mechanism appeared in pathways 25f18 (K25,18 = 0.035) to 18f17 (K18,17 = 0.101) and 25f24 (K25,24 = 0.057) to 24f17 (K24,17 = 0.013) in model 34 (see Figure 4c). As indicated in the present result, bridge-mediated terms in the electronic coupling can dominate the EET reaction, if χ for the direct term becomes small enough. This means that the EET rate can also be controlled by the design of the bridge states because the χ factor is controlled by the orientations of the donor and acceptor fragments. 3.4. Tunneling Pathway Analysis at the MO Level. Connecting the tunneling fluxes, we can find a pathway for the EET between the donor and the acceptor. In this subsection, we 10819



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Figure 5. EET pathways for model 14. Direct pathways, through-peptide indirect pathways, and through-residue indirect pathways are shown in blue, red and green colors, respectively. Arrows and numbers indicate the normalized flux, Kai,bj = pJai,bj/TIF (unitless).

discuss the components of the tunneling flux in terms of determinant contributions, which clarifies the tunneling pathway at the MO level. The analysis was applied to the long-range EET of model 14 because typical tunneling pathways provide chemical explanations of how the bridging states mediate EET. The direct pathways are shown at the center and bottom of Figure 5 (cyan arrows). The primary (direct) pathway was between the HOMOfLUMO excited determinants of the donor (fragment 1) and acceptor (fragment 4). Its relative contribution (2.44 TIF) was more than two times larger than the whole TIF (2.7 105 au). The secondary pathway was between the HOMOfLUMO determinant of the donor and the HOMO1fLUMO+1 determinant of the acceptor (“alternative direct” pathway shown in the bottom of Figure 5). Another secondary pathway was between the HOMO1fLUMO+1 determinant of the donor and the HOMOfLUMO determinant of the acceptor. Because of the phase relation of the initial- and finalstate wave functions, these secondary pathways ran from the acceptor to the donor; almost half of the contribution from the primary pathway was canceled by these two pathways. There was another pathway from the HOMO1fLUMO+1 determinant of the

donor to that of the acceptor (0.14 TIF) in the same direction as the primary pathway. These direct pathways are related to the electronic structure of the first excited state of tyrosine. The dominant components of the wave function were the HOMOfLUMO and HOMO 1fLUMO+1 determinants. The components of the direct pathway were the combination of these two excitations in the donor and acceptor. The electronic structure of the first excited state originated from the phenyl group. Figure 6a shows the HOMO1, HOMO, LUMO, and LUMO+1 orbitals of tyrosine. The two occupied MOs were closely spaced in energy, as were the two unoccupied MOs. Next, we illustrate two types of indirect pathways. The first is a through-peptide bond pathway (red arrows in Figure 5). As shown at the top of Figure 5, ππ* excitations of the peptide bonds (fragment 7) functioned as bridge determinants in the EET through-peptide tunneling flux. Compared with multistep pathways, two-step pathways through single bridge states gave major contributions to the indirect term. The result indicates that the model 14 is classified to the weak sitesite interaction limit because the magnitude of the indirect flux decreases with the 10820



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Figure 6. Localized molecular orbitals (LMOs) used for calculating the electronic coupling. (a) Tyrosine fragment (HOMO1 to LUMO+1). (b) Peptide fragment (π orbitals). (c) Phenylalanine fragment (HOMO1 to LUMO+1). Because of the localization scheme used, the LMO distributions shown above are very similar among the models.

number of the sites (see section C-2 in the Supporting Information). We note that the LMOs in the peptide bond moiety were similar to the CMOs of a truncated peptide fragment, CH3-CO-NH-CH3 (see Figure 6b), even though the LMOs were obtained using population localization. The π-orbitals in the CdO and CN bonds and the π*-orbital correspond to the HOMO1, HOMO2, and LUMO of the fragment, respectively. The second indirect pathway is a through-residue pathway. Two of the dominant pathways are shown as green arrows in Figure 5; ππ* excitations of the phenylalanine residue (fragment 2) served as the bridge determinant in the EET tunneling flux. Pathways with single bridge states gave significant contributions, as in the previous case. The main bridge configurations were the HOMO1fLUMO and HOMOfLUMO +1 excitations, which were components of a low-lying ππ* excited state of phenylalanine. The MOs of phenylalanine were similar to those of tyrosine as shown in Figure 6c.

4. CONCLUSION We have extended a method for calculating the electronic couplings for EET and ET. Singly excited determinants were used for describing exciton and charge-transfer states. We employed LMOs as the one-electron basis functions of the determinants. The LMO transformation, developed in our group, localizes CMOs within predefined local spaces, allowing the donor, acceptor, and EET pathways to be clearly defined. The Green’s function technique was employed to include the CI between the donor/acceptor and the bridge determinants up to second order perturbations. The use of determinants in the Green’s function technique represents a generalization of the method to a many-electron CI level of description. We have also extended the pathway analysis of the electronic coupling by generalizing the concept of the tunneling current to an excitedstate configuration density flux. We applied the present method to analyze the EET electronic couplings and pathways in model peptide systems. Although there are many theoretical studies on biological EET using F€orster and Dexter theories, no prior study has investigated how the protein medium contributes to the electronic coupling. The result of the present calculations is the first example of ab initio numerical evaluation of bridge-mediated EET in a peptide medium. Direct interactions between the donor and acceptor tended to dominate. Contributions from the indirect terms were more than 1 order of magnitude smaller than those of the direct terms, as shown in Figure 2. We also compared the calculated direct terms with those of the dipole model. These two

descriptions gave very similar results, as shown in Figure 3. The present results allowed numerical verification of the applicability of the F€orster theory to EET in protein media. We also note that the direct and indirect terms can be comparable in magnitude because of the orientation of the donor and acceptor. In the pathway analysis using the tunneling flux, we found that local exciton states mediate EET. The ππ* excitations of the peptide bonds and residues were important contributors. We note that these indirect terms do not necessarily enhance the total TIF; because of phase effects, some of the terms partially canceled the primary contribution from the direct term.

’ ASSOCIATED CONTENT

bS

Supporting Information. More detailed descriptions were given for (A) electronic coupling with superexchange mechanism, (B) configuration density tunneling flux analysis for excitation energy transfer, and (C) distance dependence of bridge-mediated indirect electronic coupling. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Phone: +81-75-3252746. Fax: +81-75-325-2741.

’ ACKNOWLEDGMENT This study was supported by KAKENHI (No. 21685002) from the Japan Society for the Promotion of Science (JSPS), JSTCREST. A portion of the computations were carried out at RCCS (Okazaki, Japan) and ACCMS (Kyoto University). ’ REFERENCES (1) Cheng, Y. C.; Fleming, G. R. Annu. Rev. Phys. Chem. 2009, 60, 241–262. (2) van Grondelle, R.; Novoderezhkin, V. I. Phys. Chem. Chem. Phys. 2006, 8, 793–807. (3) van Grondelle, R.; Dekker, J. P.; Gillbro, T.; Sundstr€om, V. Biochim. Biophys. Acta, Bioenerg. 1994, 1187, 1–65. (4) Hu, X. C.; Ritz, T.; Damjanovic, A.; Autenrieth, F.; Schulten, K. Q. Rev. Biophys. 2002, 35, 1–62. (5) Renger, T.; May, V.; Kuhn, O. Phys. Rep. 2001, 343, 138–254. (6) Laquai, F.; Park, Y. S.; Kim, J. J.; Basche, T. Macromol. Rapid Commun. 2009, 30, 1203–1231. (7) Baldo, M. A.; Thompson, M. E.; Forrest, S. R. Nature 2000, 403, 750–753. (8) Liu, Y. X.; Summers, M. A.; Edder, C.; Frechet, J. M. J.; McGehee, M. D. Adv. Mater. 2005, 17, 2960–2964. 10821


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Bridge-Mediated Excitation Energy Transfer Pathways through Protein

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