Theoretical Prediction of TripletâTriplet Energy Transfer Rates in a

Article pubs.acs.org/JPCC

Theoretical Prediction of Triplet−Triplet Energy Transfer Rates in a Benzophenone−Fluorene−Naphthalene System Yubing Si, Wanzhen Liang, and Yi Zhao* State Key Laboratory for Physical Chemistry of Solid Surfaces, Fujian Provincial Key Lab of Theoretical and Computational Chemistry, and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005, China S Supporting Information *

ABSTRACT: Triplet−triplet energy transfer in benzophenone−fluorene and benzophenone−fluorene−naphthalene molecules is theoretically investigated by using the rate theories and electronic structure calculations established for electron transfer. From the calculated electronic couplings for the single-step tunneling and multistep hopping pathways of the energy transfer from the donor benzophenone to the acceptor naphthalene, it is found that the tunneling comes from the direct electronic couplings between the donor and acceptor states, other than the coupling via the virtual bridge state in the conventional superexchange mechanism. The mode-specific reorganization energy calculations reveal that only the several highfrequency modes dominate the energy transfer, leading to an important nuclear tunneling effect. Succeedingly, with use of the obtained parameters, Fermi’s golden rule predicts the consistent energy transfer rates with experimental ones. the Dexter mechanism.11 It is also named as the overlap or collision mechanism occurring when the molecule D* and the molecule A are close enough. If the electron clouds of the donor and acceptor molecules sufficiently overlap each other, an exciton could diffusively hop from one molecule to the next without spin flip. On the basis of these mechanisms, an amount of approaches have been proposed to calculate the electronic coupling between D*A and DA* states.12−15 Essentially, Förster electronic coupling corresponds to a Coulombic interaction, and additional exchange integral contributions should be incorporated in Dexter electronic coupling. In TTET, however, both spin and energy are exchanged between the initial state 3DA and the final state D3A, although the overall spin is conserved. The Coulombic interaction thus cannot be applied to understand the transfer mechanism because of spin-forbidden. It is now clear that TTET is promoted by interactions related to the orbital overlap between the initial and final diabatic states. Scholes’ group16 has suggested a general route to calculate the electronic coupling between the donor and acceptor states, which involves the effect of a given initial diabatic state that electronically couples with more than one final state. The electronic couplings for both the singlet−singlet and triplet−triplet intermolecular energy transfer are expressed in a concise way. A simpler and clearer formula based on Scholes’ approach has also been proposed by Shi et al. recently.17 These approaches are suitable for both short- and long-range energy transfer, and especially,

1. INTRODUCTION Recently, Vura-Weis et al.1 have measured the rates of molecule triplet−triplet energy transfer (TTET) from a benzophenone (Bp) donor to a naphthalene (Nap) acceptor via fluorene (F) bridges. They observed the crossover from single-step tunneling to multistep hopping as the bridge length increases. Their phosphorescence measurements also showed that the triplet-state energies from the bridges to the acceptor are downhill, manifesting that the bridges are energetically accessible during the energy transfer. The experimental results between the strong distance dependence of tunneling and weak distance dependence of hopping are proved to be useful in understanding the energy transfer pathways in photovoltaic devices with use of organic π stack molecules, such as the lightemitting diodes.2−4 In the present paper, we theoretically model the TTET experiment of Vura-Weis et al.1 by combining electronic structure calculations and rate theories to quantitatively predict the TTET rates. The main purpose is to find suitable computational approaches for the investigation of the energy transfer in general π stack organic molecules. An easy way to understand energy transfer starts from a singlet exciton model of a pair of molecules or molecular fragments, where an excited donor (D*) molecule (exciton) transfers its energy to an acceptor (A) which in turn is promoted into an excited state (A*), i.e., D*A → DA* (for instance, see refs 5−8 and references therein). There are two well-known mechanisms to explain energy transfer processes. One is the Fö rster mechanism,9,10 which is identified as a nonradiative resonance excitation energy transfer between a pair of molecules over distances much greater than collision diameters. The other is © 2012 American Chemical Society

Received: April 17, 2012 Revised: May 20, 2012 Published: May 21, 2012 12499

dx.doi.org/10.1021/jp303705d | J. Phys. Chem. C 2012, 116, 12499−12507

The Journal of Physical Chemistry C

Article

applicable or not for the present system. We will clarify this problem in the present work.

they also incorporate the charge transfer exciton contribution which may be important for symmetric dimer systems.18,19 In the present work, the charge transfer exciton may be neglected because asymmetric systems are considered. In this case, there are several easier ways to find diabatic states.20−26 Here, we use the constrained density functional theory (CDFT) proposed by Voorhis’ group,14 which provides a direct route to diabatic electronic states. The electronic coupling is succeedingly obtained from the orbital overlap method. For the quantitative prediction of the TTET rate, one needs to know the reorganization energy and driving force besides the electronic coupling. Once the initial and final diabatic states are defined, the driving force can be obtained from the energy difference of the optimized two diabatic states. The reorganization energy is related to the nuclear relaxation property during the course of energy transfer. The four-point model,27 the mode-specific model,28−30 or a method based on molecular dynamics simulation31 proposed for electron transfer (ET) can be applied. With the use of these parameters from quantum calculations, the TTET rate is ready to be calculated from the available rate theories (for instance, see ref 32 and references therein). Following this line, Voityuk33 and we34 have separately investigated the TTET rate between two fluorene molecules which act as the bridges in Vura-Weis’s experiment.1 Although the experimental rate is not available to compare with, theoretical rates have an order similar to the experimental total rate from the Bp donor to the Nap acceptor. In this work, we consider TTET from the Bp donor to the F bridge as well as to the Nap acceptor to confirm the validity of the theoretical methods. The TTET from the Bp to F molecules involves two asymmetric diabatic states. The corresponding rate is easily calculated from the two-state model. For the Bp−F−Nap system, however, three or more diabatic states are involved. For such a kind of donor−bridge−acceptor (D−B−A) system, TTET essentially has two transfer channels, the tunneling (superexchange) and sequential hopping. These two mechanisms are well-known in the ET process and have been extensively investigated by many analytical and numerical approaches to understand the factors of governing their competition.35−38 In the superexchange channel, the electron tunnels quantum mechanically from the donor to the acceptor through the energetically well-isolated bridges (3DBA → DB3A), associated with a characteristic exponentially decreasing yield with increasing bridge length. While in the sequential hopping mechanism, energy is temporarily localized on the bridges and a chemical intermediate produced (3DBA → D3BA → DB3A), and the rate is weakly distance dependent. The competition of two channels is explained in terms of both energies and times. If the thermal energy is of the order of the energy gap between the eigenvalues of the bridge and the donor, one expects to observe either resonant tunneling for weak coupling with environment or sequential hopping for strong coupling. On the other hand, if the time of the electron staying on the bridges is shorter than the period of the environmental vibrations or polarizations that provide electron localization, the superexchange is preferred; otherwise, the hopping procedure is observed. For the present systems, however, the energies of bridge F are lower than the donor Bp.1 Therefore, the bridges are always energetically accessible during the course of energy transfer. It is not clear whether the conventional description for the competition of the two channels with the energetically well-isolated bridges is still

2. COMPUTATIONAL METHODS 2.1. Rate Expression. The rate calculations for both the energy and electron transfers that occurred in two electronic states are essentially the same, although their transfer mechanisms are explicitly different. The two-state Hamiltonian can be modeled as follows 2

H=

∑ |i⟩Hij⟨j| (1)

i,j=1

Here, |i⟩ represents the electronic states. H12(≡ H21) is the electronic coupling, which is conventionally assumed to be independent of nuclear motions (Condon approximation). The diagonal Hii correspond to the energies of the diabatic states. As the nuclear motions are considered as a collection of harmonic oscillators, they are written as H11 =

∑ k

Pk2 1 + ωk2Q k2 2 2

H22 = ΔG +

∑ k

Pk2 1 + ωk2(Q k − Q 0k)2 2 2

(2)

under mass-reduced coordinates. Here, ΔG is the driving force. Pk and Qk are the momentum and coordinate of the k-th mode, respectively. As the frequency ωk in the two states is assumed to be the same, the total reorganization energy is then determined from the coordinate shift Q0k by λ = ∑kλk = ∑k(1/2)ω2k Q20k, where λk is defined as the mode-specific reorganization energy. Once eq 2 is constructed from electronic structure calculations, the radiationless rate between the two states can be readily calculated.32 Especially, as the electronic coupling H12 is weak, the transfer rate has an analytical solution. For one effective mode with frequency ω, the rate is given by Fermi’s golden rule (FGR)39,40 k=

2 2πH12 ⎛ n + 1 ⎞ p /2 ⎜ ⎟ exp[−S(2n + 1)] 2 ℏω ⎝ n ⎠

Ip[2S(n(n + 1))1/2 ]

(3)

where S(= λ/ω) is the Huang−Rhys factor; T is the temperature; n = (eℏω/kBT − 1)−1; p = ΔG/ℏω; and Ip is the modified Bessel function. The effective frequency is related to those of multimodes by41 1 ω 2 = ∑ ωi2λi λ i (4) In the high-temperature limit (ℏω ≪ kBT), eq 4 becomes the well-known Marcus formula42,43 k=

⎡ (ΔG + λ)2 ⎤ π 1 2 H12 exp⎢ − ⎥ 4λkBT ⎦ ℏ λkBT ⎣

(5)

To get the rate for concrete systems, it therefore needs to know the electronic coupling, driving force, and mode-specific reorganization energy. In the following, we outline the electronic structure methods for the calculations of these quantities. 2.2. Geometries. Before the calculations of parameters used in the TTET rate, one has to find the correct diabatic 12500



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Figure 1. Structures of the Bp−F and Bp−F−Nap.

states of donor (3DBA) and acceptor (DB3A). It may be possible to optimize these triplet states of the D−B−A supermolecule starting from a careful guess of initial geometry with conventional electronic structure methods, such as Hartree−Fock (HF) and density functional theory (DFT). However, it is known that DFT with popular B3LYP functional44 provides a poor description of the π−π stacking systems due to the known problems on the nonlocality and asymptotic behavior of available density functionals,45−47 and the optimized geometries are not always corresponding to the correct donor and acceptor states; i.e., most of the spin density is not located on the donor or acceptor segment but delocalized over the whole DBA molecule. To find the suitable diabatic states, here, we use CDFT with B3LYP, M06-2X,48 and ωB97X-D49 functionals and the 6-31G* basis set in the Qchem software package.50 In the CDFT calculations, the Becke weights51 are used in the constrained population analysis to make sure that the spin density locals on the donor, bridge, and acceptor, respectively. 2.3. Reorganization Energy. The reorganization energy in TTET is the energy cost due to geometry modifications to go from a triplet to a singlet state and vice versa. Therefore, it is not sensitive to the relative positions of the donor and acceptor molecules. An easy way to calculate the reorganization energy is the four-point technique proposed by Nelsen,27 which has been identified as a reliable tool by our previous study on the ET process in organic radicals.30,52 Here, we use it for the energy transfer. For the neutral TTET, similar to the ET process, the molecule can be split into the neutral triplet donor and singlet acceptor parts which are calculated separately or can be taken as a whole to calculate. If we use opt and fc to represent the optimized states and Franck−Condon excited states, respectively, and the spin multiplicity is labeled as the superscript, the reorganization energy is then given by

λ = λ1 + λ 2 = [E(3D fc) − E(3Dopt )] + [E(1A fc) − E(1A opt )]

(6)

To calculate the mode-specific reorganization energy, we first make normal-mode analysis in both optimized donor and acceptor states in the same Cartesian coordinates. The shift Q0k of the k-th mode between two optimized geometries is calculated by the matrix transformation techniques.32,53−58 The total reorganization energy is then calculated by λ = ∑k(1/ 2)ω2k Q20k, and the effective frequency is readily calculated by eq 4. 2.4. Electronic Coupling. The electronic coupling is commonly calculated from the adiabatic and diabatic representation. For instance, the fragment excitation difference (FED)12 and the fragment spin difference (FSD) approaches13 are based on the adiabatic representation. In the adiabatic-based approaches, one has to carefully select the proper excited states and the transition densities from CIS/TDDFT. On the other hand, the CDFT approach starting from the diabatic representation constrains the spin or/and charge to a special regime of the supermolecule to generate the diabatic states.59 The electronic coupling is related to the overlap of the wave functions of two diabatic states. In this paper, the CDFT method is adopted to calculate the electronic coupling of TTET, and other methods are also used for the purpose of comparison. In the application of the CDFT approach, as already done by Yeganeh and Voorhis for the TTET process,51 we constrain the two parallel electronic spins in the donor and acceptor molecule, respectively, and the electronic coupling is then calculated by 1

Hda = 12501

Vda − 2 (Ed + Ea)Sda 2 1 − Sda

(7)



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where Ed and Ea are the diabatic energies of the donor and acceptor states, respectively, and Vda and Sda are defined as follows Vda = ⟨3DBA|Ĥ |DB3A⟩

(8)

Sda = ⟨3DBA|DB3A⟩

(9)

the singlet ground state). However, at the B3LYP level, the obtained spectral energy (2.34 eV) is much lower than the experimental one (3.04 eV),1 and theoretical spectra cannot be further improved by using better functionals, such as the hybrid meta functionals (M06-2X) and long-range corrected functionals (ωB97X-D) and the larger basis sets and larger grid. After the analysis of the spin density and charge distribution, we find that too many (Mulliken) charges (−0.33 e) are localized on the oxygen in Bp. This result is incorrect according to the previous investigation which shows that about one-third of the triplet spin density should be localized on the benzene rings in Bp.62,63 Therefore, we use CDFT with the constraint that the two benzene rings have one-third spin density and about −0.1 e localized on the oxygen atom.64 The obtained phosphorescence emitted from the 3Bp--F and Bp−3F states is 2.89 and 2.53 eV at the B3LYP/6-31G* level, respectively, close to the experiments of 3.04 and 2.78 eV.1 Table 1 lists the phosphorescence and several corresponding geometrical parameters as well as experimental values. It clearly

where | DBA⟩ and |DB A⟩ correspond to the triplet-state wave functions of the donor and acceptor states, respectively. The above “two-state” model is also easily applied to calculate the electronic couplings between the donor and bridge, bridge and bridge, and bridge and acceptor states, by choosing the corresponding diabatic states. For instance, the electronic coupling of the donor and bridge states can be calculated by choosing the diabatic 3DBA and D3BA states with use of eq 7. However, as the bridge state D3BA is incorporated, the superexchange transfer from the donor state to the acceptor state has to be involved. Starting from the second-order perturbation in the rate theory, it is easily shown that the electronic coupling between donor and acceptor states becomes 3

H̅da = Hda +

3

HdbHba Ed − E b

Table 1. Geometrical Properties of the Diabatic 3Bp−F and Bp−3F States

(10)

3

where Hda is given by eq 7; Hdb and Hba represent the electronic couplings between the donor and bridge states and the bridge and acceptor states, respectively; and Ed and Eb are the energies of donor state 3DBA and bridge state D3BA, respectively. In eq 10, the first term Hda represents the direct coupling between the donor 3DBA and acceptor DB3A states, and the second term comes from the second-order perturbation via the bridge state D3BA. This second term is nothing but the well-known superexchange expression of electronic coupling.40,60 It now becomes clear to distinguish the hopping and superexchange contributions to the energy transfer by comparing the values of these two terms, especially for short bridges where the direct coupling may not be small.

Bp−F

Bp−3F a

exptl B3LYPa exptl B3LYPa

C−O (Å)

θ°

ωC−O (cm−1)

T1 → S0 (eV)

1.330 1.319 1.231 1.227

30 26 33 29

1222 1260 1665 1667

3.04 2.89 2.78 2.53

The frequencies obtained from B3LYP/6-31G* are scaled by 0.961.65

shows that the CO bond length in the triplet (3Bp−F) state of Bp is explicitly elongated compared to that in its singlet (Bp−3F) state, and their values are also not far from experimental ones. This can be understood by that the excitation of an electron from the singlet Bp−F state to the triplet 3Bp−F state comes from the n orbital to π* orbital (n → π*). This excitation weakens the CO bonding and elongates the bond length from 1.227 to 1.319 Å. The previous investigations also predict the same tendency.66−68 The parameter θ, which is defined as the twist angle between the phenyl rings, becomes larger in the Bp−3F state than that in the 3 Bp−F state. This tendency is again consistent with the experimental measurements. The explicit deviation in the 3Bp− F may come from the delocalization of the carbonyl π* into the phenyl ring π* orbital in Bp which causes the decrease of the steric interaction of phenyl rings.69−71 Another interesting property is that the carbonyl stretching frequency increases from 1260 cm−1 in the 3Bp−F state to 1667 cm−1 in the Bp−3F state because of the different equilibrium bond lengths. These data agree with the experimental IR spectra data 1222 and 1665 cm−1 well.72,73 The frequency change in the different spin state of oxygen has also been detected by Gorman74 and Darmanyan.75 To further improve the computational accuracy, we also test the functionals M06-2X and ωB97X-D. Although they can indeed predict the more accurate phosphorescence data in 3 Bp−F and Bp−3F, they predict the larger frequency of the C− O oscillator and the larger coordinate shift after energy transfer, which will make the mode-specific reorganization energy larger than the true data. Moreover, the geometries are hard to converge. In addition, the larger basis set which includes the diffuse function, such as the 6-31+G**, does not give the better results (see Table S1, Supporting Information).59 In the

3. RESULTS AND DISCUSSION 3.1. TTET in the Bp−F Molecule. The energy transfer in the Bp−F molecule is the initial step in the Bp−F−Nap molecule. The molecular geometries are shown in Figure 1. We first focus on the TTET in the Bp−F molecule. Bp is a very good donor for the triplet energy transfer because its singlet excitation energy can almost completely transfer into its triplet state after photoexcitation.61 This triplet energy succeedingly transfers into the F molecule. Thus, one may model the TTET between Bp and F as a radiationless process occurring in two diabatic 3Bp−F and Bp−3F states where the triplet excitations are localized on the Bp and F molecule, respectively. It is known that the conventional electronic structure calculations start from the adiabatic representation and cannot be used straightforwardly to determine the diabatic geometries of molecules. Similar to ET, however, one expects that the stable diabatic geometries should be very close to the adiabatic ones. By using the DFT methods and 6-31G* basis set after carefully choosing initial geometric guesses (here, the CDFT is used to get initial geometries), indeed, we have obtained such the optimized diabatic states where the triplet spin densities are localized on Bp and F molecules, respectively. We further calculate the phosphorescence spectrum of the 3Bp−F state (the vertical emission energy from the optimized triplet state to 12502



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modes with frequencies of 1311 and 1662 cm−1 have very large reorganization energies compared to other modes. The mode with 1697 cm−1 corresponds to the CC bond stretching motion in the F dimer molecule, which has been reported in our previous investigation in the TTET between the F dimer.34 The mode with 1311 cm−1 is the CO stretching motion in the Bp molecule, and a similar behavior has been predicted by Closs et al.80 Thus, we expect that these two modes mainly control the energy transfer process. To get the TTET rate, the rest task is to obtain the electronic coupling. On the basis of the optimized diabatic geometry from the CDFT, we calculate the electronic coupling in NW-Chem 6.0.81 The obtained results are 2.4 and 5.7 meV at the optimized geometries of 3Bp−F and Bp−3F states, respectively. For the purpose of comparison, the FSD approach based on the adiabatic state is also used, and the predicted electronic coupling is 4.3 meV at the singlet geometry of the Bp−F state from the Hartree−Fock configuration-interaction singles (HFCIS) method. Both methods predict the close electronic couplings. In generality, DFT and HF predict different electronic couplings.30 The consistency here can be understood from the variational principle of CDFT. When Beck’s atomic partitioning scheme82 is added in the exchange part of DFT, which brings the CDFT “shift” to the HF formalism, the CDFT thus becomes a constrained HF (CHF) method.51 In other words, the CDFT is more like HF than the regular DFT functionals. Thus, one expects that the electronic coupling predicted from CDFT may be similar to that from the HF method. It is noted that the electronic couplings at the optimized donor and acceptor states are explicitly different. In this case, the electronic coupling at the molecular geometry with the degenerated energies of the donor and acceptor states, i.e., at the crossing point of diabatic potential curves of the donor and acceptor states, dominates the energy transfer. To find such a kind of specific geometry, a simple technique of linear reaction coordinate (LRC)83,84 may be used to describe the energy transfer pathway and find the crossing point. In LRC, a onedimensional reaction coordinate ξ is introduced to determine the molecular geometry by

following calculations, we thus choose B3LYP/6-31G* to obtain the parameters of the states. The above analysis convinces us that the obtained diabatic states are reasonable. We can then calculate the driving force, reorganization energy, and electronic coupling based on these states. The driving force can be straightforwardly obtained from the energy difference between the optimized 3Bp−F and Bp−3F states. With the thermal dynamic correction, the calculated free energy gap ΔG is −0.25 eV, consistent with the experiment of Vura-Weis et al.1 which shows that the TTET is energetically downhill from the donor to the bridge. Now, we focus on the reorganization energy. Similar to the ET process, the total reorganization energy in TTET should consist of an inner and outer part, λV and λS. λV corresponds to the change of the intramolecular geometry of the donor and acceptor states, while λS comes from the solvent response. Toward this goal, one can employ the QM/MM protocol for TTET simulations and use it to evaluate the influence of solvent environment.76 Unlike the ET process, TTET does not involve a major redistribution of the charges between the donor and the acceptor, and the dipole moments of the individual moieties may be unchanged. It has been shown that a 0.88 D change in dipole moment accompanies the transition from T1 to S0 in Bp.77,78 In the present calculations, the change is about 2.45 D for the whole molecule at the B3LYP level (the dipole moments are 0.90 and 3.35 D for 3Bp−F and Bp−3F, respectively). Therefore, the corresponding solvent reorganization energy should be very small (0.1 eV or less79). In the following, we only consider the calculations of λV. λV can be calculated by using the four-point method. The obtained results are 0.84 eV as the Bp and F molecules are treated independently and 0.82 eV when they are considered together. Both results are close to each other, manifesting that the interaction between the Bp and F molecules is not strong. Although the four-point method is easy to implement, it cannot reveal frequency-dependent reorganization energies which is important for a detailed understanding of the energy transfer process. Therefore, we calculate the mode-specific reorganization energies, and the results are shown in Figure 2. Summing

Q i = ξQ id + (1 − ξ)Q ia

(11)

Here, Qi refers to the i-th internal coordinate (bond length, bond angle, or dihedral). Qdi and Qai refer to those at the optimized geometries of the donor and acceptor states, respectively. As ξ switches from 1 to 0, the molecular geometries change from optimized donor to the acceptor geometry. The diabatic curve crossing can be found along the ξ coordinate. For instance, ξ should be 0.5 for the symmetric energy transfer.30 Figure 3 displays the electronic coupling as well as two diabatic potentials along the ξ. It is seen that electronic couplings show rapid change with respect to ξ. The main change comes from the elongation of CO in Bp (from 1.23 to 1.31 Å), and other parameters such as θ and the dihedral angle between Bp and F planes do not change too much. Therefore, the electronic couplings nearly have constant values of 3.9 meV around the crossing point (ξ = 0.65). This value is thus reasonable in the calculation of the TTET rate. Now, it becomes possible to calculate the TTET rate. In the numerical calculation, the reorganization energy is 0.86 eV; the driving force is −0.25 eV; and the electronic coupling is 3.9 meV. From the electronic coupling and the reorganization

Figure 2. Reorganization energy components as a function of frequency at the B3LYP/6-31G* level in the Bp−F molecule.

over these reorganization energies, we obtain the total inner reorganization energy of 0.86 eV. This result is consistent with those from the four-point method. Therefore, the harmonic approximation of nuclear motions is suitable for the Bp−F molecule during the energy transfer. Figure 2 obviously shows that the reorganization energies dominantly come from the several modes, and there is little contribution in the high frequency region (>1750 cm−1) where most modes correspond to C−H stretching vibrational motions. More interestingly, two 12503



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the 3Bp−F−Nap (D), Bp−3F−Nap (B), and Bp−F−3Nap (A) states. The corresponding phosphorescent spectra are 2.89, 2.54, and 2.37 eV, respectively. It is noted that these phosphorescent spectra for the D and B states agree with those calculated for the 3Bp−F and Bp−3F states and experimental measurements for the isolated 3Bp and 3F molecules very well, manifesting that the Nap molecule does not affect the D and B diabatic states too much and isolated 3 Bp and 3F molecules dominantly determine the triplet D and B states, respectively. However, the property of the triplet A state may be significantly different from that of the isolated 3Nap molecule because its phosphorescence is explicitly different from the isolated 3Nap molecule for which the calculated spectrum is 2.49 eV, in agreement with the experimental value of 2.64 eV.86 On the basis of these diabatic states, we calculate the parameters for the control of TTET rates. Table 2 lists the Table 2. Calculated Parameters for the Diabatic States 3Bp− F−Nap (D), Bp−3F−Nap (B), and Bp−F−3Nap (A)

Figure 3. Diabatic potentials and the electronic couplings along the LRC in the Bp−F molecule at the B3LYP/6-31G* level.

energy, we find that 4V/λ is much smaller than 1, manifesting that FGR is very suitable to estimate the rate.85 To use the FGR with an effective mode, we calculate the effective frequency from the mode-specific reorganization energy, and the result is 1409 cm−1. The leading TTET rate from eq 3 is 4.7 × 1010 s−1 (21 ps) at room temperature. It is noted that this effective frequency is high enough to make the quantum effect explicit at room temperature. Indeed, the well-known Marcus formula from the high-temperature approximation of the FGR predicts the rate constant is 5.2 × 109 s−1 (239 ps), which is 10 times smaller than that from the FGR. Comparing to the experimental injection time (156 ps),1 the theoretical value (FGR) is 7 times larger, which is reasonable because the present calculation does not incorporate the acceptor molecule and solvent effect. 3.2. TTET in Bp−F−Nap. As the Nap molecule is attached to the bridge F molecule, the energy transfer processes from the donor Bp to acceptor Nap molecules follow the hopping and superexchange pathways, corresponding to the second-order and higher-order perturbation of the electronic coupling, respectively. The hopping pathways involve 3Bp−F−Nap → Bp−3F−Nap → Bp−F−3Nap and 3Bp−F−Nap → Bp− F−3Nap, and the superexchange pathway is 3Bp−F−Nap → Bp−F−3Nap. Although the superexchange pathway is the same as the second one in the hopping process, their mechanisms are completely different. In the hopping model, there is a direct electronic coupling between 3Bp−F−Nap and Bp−F−3Nap states, and the rate corresponds to the second-order perturbation of this coupling, whereas the rate in the superexchange is proportional to the second-order perturbation of both electronic couplings between the 3Bp−F−Nap and Bp−3F−Nap states and the Bp−3F−Nap and Bp−F−3Nap states (see eq 10). To determine the rates along individual pathways, similar to the previous section for the Bp−F supermolecule, we need to know the driving forces, reorganization energies, and electronic couplings between the diabatic states. The diabatic states are determined by using the CDFT with B3LYP/6-31G*. It is found that the triplet spin density can be well localized on the Bp, F, and Nap molecules, respectively, for

parametersa

D→ B

B→A

D→A

ΔG (eV) λ1 (eV) λ2 (eV) Hij (meV) ω (cm−1) kM(T) (1010/s) kF(T) (1010/s) kexpt1 (1010/s)

−0.24 0.83 0.94 3.5 1414 0.14 2.5 0.64

−0.30 0.86 0.89 5.1 1368 1.1 8.9 ≫0.64

−0.55 0.77 0.82 1.7 1403 2.3 2.6 0.95

λ1 and λ2 are from the four-point model and the mode-specific method, respectively. kM(T) and kF(T) correspond to the rates from the Marcus formula and FGR, respectively. a

corresponding results. It is seen that the driving forces for the D → B → A are downhill, which is consistent with experimental prediction. Interestingly, the reorganization energies for D → B, B → A, and D → A are close to each other based on the fourpoint model. However, the values from the mode-specific reorganizations are slightly larger than those from the fourpoint method, especially in the D → B process. These discrepancies may be explained by the harmonic approximation of nuclear motions. Indeed, we find that the bond length of the CO bond changes from 1.227 to 1.319 Å after energy transfer. Such a large change may cause the anharmonic effect. Table 2 also displays that the effective frequencies in three pathways are high (>1300 cm−1), manifesting that the nuclear tunneling effect in the rate becomes important. Those high frequencies of the effective modes can be understood from the mode-specific reorganization energies shown in Figure 4. Although several modes dominate the total reorganization energy in all three pathways, the corresponding modes are slightly different in different pathways. In the D → B process, the two modes from the motions of the CO stretching in Bp and CC stretching in F play an important role, similar to the process of 3Bp−F → Bp−3F. In the B → A process, although the CC stretching in F is still an important mode, two other modes in Nap appear with the slightly lower frequencies than the CO mode in Bp. In the D → A, however, many modes with low frequencies have a small contribution to the total reorganization energy, and the CC and C−O stretching motions dominate the contribution. Therefore, one expects that 12504



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Figure 5. Diabatic potentials and electronic couplings with respect to the LRC. The green and magenta lines represent the direct and superexchange electronic couplings, respectively.

Figure 4. Reorganization energy components as a function of frequency at the B3LYP/6-31G* level in the Bp−F−Nap system.

CC and C−O stretching modes are extremely important in the energy transfer. For the electronic couplings, we find that they are explicitly dependent on molecular geometries. These non-Condon effects may have the dynamic contribution to the energy transfer rates.39,58 However, it is noted that the TTET rate is in the inverse of picoseconds, and it should be faster than the motion of low-frequency modes which have a significant non-Condon effect. Therefore, it is safe to use the average electronic couplings. The values of electronic couplings in Table 2 are from the average along the LRC. It is interesting to note that the direct coupling from the D to A states is nonzero, and it corresponds to the first term in eq 10. To calculate the superexchange electronic coupling (the second term in eq 10), we have to find the supermolecular geometry at which the energies of the D and A states are degenerated.87 To do so, we calculate the three diabatic potentials along the LRC. The results and the corresponding electronic couplings are shown in Figure 5. The superexchange electronic coupling is 1.9 × 10−5 eV (at ξ = 0.85 where the donor and acceptor energies are equal). Compared with the direct coupling (1.7 × 10−3 eV), this superexchange coupling is two orders smaller. With use of the obtained parameters, we calculate the TTET rates from both the FGR and Marcus formulas, and the results are also listed in Table 2. The corresponding schematic TTET pathways are shown in Figure 6 with solid lines. For the purpose of comparison, we also display the rate for the sole superexchange pathway in Figure 6.

Figure 6. Schematic pathways for TTET processes. The optimized triplet energy levels of 3Bp−F−Nap, Bp−3F−Nap, and Bp−F−3Nap are determined from the phosphorescence spectroscopy at the B3LYP/6-31G* level. The TTET rates which are calculated by FGR are shown as solid (green and blue) lines, and the superexchange rate from Bp to Nap is also shown as a dashed red line.

Now, it becomes possible to compare the calculated rates with the experimental values.1 In the three pathways, the rate from the FGR is the largest for B → A, and the rate of D → A is larger than that of D → B. This tendency is consistent with the experimental one. Furthermore, the calculated rates are also close to the experimental values. For instance, the experimental rate for the D → B is 6.4 × 109 s−1 (156 ps), which is only 3 times smaller than the calculated rate. In the rate calculations, we have neglected the solvent reorganization energy. Although this reorganization energy should be small, it may explicitly lower the effective frequency. If this effect is considered, the FGR rate should be much closer to the experimental one. It is interesting to note that the rates from the Marcus hightemperature formula are explicitly smaller than those from the FGR, especially for D → B. It can be explained by the nuclear tunneling effect. It is known that the potential barrier for TTET becomes small for negative driving force in the Marcus normal regime. Since D → B has less negative driving force than B → 12505



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A, it has a higher barrier. At a given temperature and effective frequency, the tunneling effect becomes more important for a higher barrier. Indeed, the FGR rate is 18 times faster than the Marcus rate for D → B, whereas it becomes eight times faster for B → A. This can be further confirmed by the TTET rates from D → A, where both FGR and Marcus rates are nearly the same because the barrier becomes very small in this case (at λ = −ΔG, the barrier disappears).

AUTHOR INFORMATION

Corresponding Author

*Fax: +86-0592-2183047; E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We express many thanks to Professor Alexander A. Voityuk for his helpful discussions. This work is partially supported by the National Science Foundation of China (Grant Nos. 20833004, 20833003, 21073146, and 21133007) and the National Basic Research Program of China (Grant no. 2011CB808501).

4. CONCLUDING REMARKS We have theoretically investigated the triplet−triplet energy transfer (TTET) in benzophenone−fluorene (D → B) and benzophenone−fluorene−naphthalene (D → B → A) which have been measured experimentally by Vura-Weis et al.1 It is found that the TTET rates can be completely predicted from the rate theories and electronic structure calculations commonly used in the study of electron transfer. In the electron structure calculations, the construction of the accurate diabatic states is extremely important for the calculations of the driving force, reorganization energy, and electronic coupling, the key parameters for the control of energy transfer. We have used constrained density functional theory (CDFT) to find these diabatic states. Although the CDFT is easily implemented, one should carefully choose the constraint conditions. For instance, one cannot constrain the triplet spin on the whole benzophenone molecule because of the property of the oxygen atom in it. From the calculation of reorganization energies, we find that only the several high-frequency modes dominate the TTET, leading to the important nuclear tunneling effect. From the electronic couplings, more interesting results are obtained. First, the electronic couplings for the energy transfer in the systems of the benzophenone−fluorene and benzophenone− fluorene−naphthalene are very sensitive to the molecular geometries, which has been confirmed by the linear reaction coordinate dependence of the electronic coupling. Second, the direct electronic coupling between the donor and acceptor cannot be ignored for the D−B−A system with the pathway of downhill energy. In the present calculations, the electronic couplings for the D → B and D → A only have two times difference. Finally, it should be addressed that the conventional superexchange rate in the present system is very small compared to the rate from the direct electronic coupling. Therefore, the single-step tunneling mentioned in the experiment should correspond to the rate from the direct electronic coupling rather than the superexchange mechanism. Although we have obtained the consistent TTET rates with the experimental ones, we limit the investigation on the systems with a single bridge. As multiple bridges are involved, one has to consider the coherence motion between bridges. The twostate perturbation rate approaches used in the present paper may not be suitable for the investigation of the energy transfer dynamics.

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REFERENCES

(1) Vura-Weis, J.; Abdelwahed, S. H.; Shukla, R.; Rathore, R.; Ratner, M. A.; Wasielewski, M. R. Science 2010, 328, 1547. (2) Hanss, D.; Walther, M. E.; Wenger, O. S. Chem. Commum. 2010, 46, 7034. (3) Sun, D. L.; Rosokha, S. V.; Lindeman, S. V.; Kochi, J. K. J. Am. Chem. Soc. 2003, 125, 15950. (4) Hughes, G.; Bryce, M. R. J. Mater. Chem. 2005, 15, 94. (5) Scholes, G. D. Annu. Rev. Phys. Chem. 2003, 54, 57. (6) Andrews, D. L.; Curutchet, C.; Scholes, G. D. Laser Photonics Rev. 2011, 1, 114. (7) Olaya-Castro, A.; Scholes, G. D. Int. Rev. Phys. Chem. 2011, 1, 49. (8) Albinsson, B.; Mårtensson, J. Phys. Chem. Chem. Phys. 2010, 12, 7338. (9) Föster, T. Ann. Phys. 1948, 437, 55. (10) Föster, T. Discuss. Faraday Soc. 1959, 27, 7. (11) Dexter, D. L. J. Chem. Phys. 1959, 21, 836. (12) Hsu, C. P.; You, Z. Q.; Chen, H. C. J. Phys. Chem. C 2008, 112, 1204. (13) You, Z. Q.; Hsu, C. P. J. Chem. Phys. 2010, 133, 074105. (14) Voorhis, T. V.; Kowalczyk, T.; Kaduk, B.; Wang, L. P.; Cheng, C. L.; Wu, Q. Annu. Rev. Phys. Chem. 2010, 61, 149. (15) Cave, R. J.; Newton, M. D. Chem. Phys. Lett. 1996, 249, 15. (16) Scholes, G. D.; Harcourt, R. D.; Ghiggino, K. P. J. Chem. Phys. 1995, 102, 9574. (17) Shi, B.; Gao, F.; Liang, W. Z. Chem. Phys. 2012, 394, 56. (18) Gao, F.; Liang, W. Z.; Zhao, Y. J. Phys. Chem. A 2009, 113, 12847. (19) Pan, F.; Gao, F.; Liang, W. Z.; Zhao, Y. J. Phys. Chem. B 2009, 113, 14581. (20) Subtonik, J. E.; Vura-Weis, J.; Sodt, A. J.; Ratner, M. A. J. Phys. Chem. A 2010, 114, 8665. (21) Ohta, K.; Closs, G, L.; Morokuma, K.; Green, N. J. J. Am. Chem. Soc. 1986, 108, 1319. (22) Broo, A.; Larsson, S. Chem. Phys. 1990, 148, 103. (23) Farazdel, A.; Dupuis, M.; Clementi, E.; Aviram, A. J. Am. Chem. Soc. 1990, 112, 4206. (24) Zhang, L. Y.; Friesner, R. A.; Murphy, R. B. J. Chem. Phys. 1997, 107, 450. (25) Subotnik, J. E.; Yeganeh, S.; Cave, R. J.; Ratner, M. A. J. Chem. Phys. 2008, 129, 244101. (26) Vura-Weis, J.; Wasielewski, M.; Newton, M. D.; Subotnik, J. E. J. Phys. Chem. C 2010, 114, 20449. (27) Nelsen, S. F.; Blackstock, S. C.; Kim, Y. J. Am. Chem. Soc. 1987, 109, 677. (28) Bredás, J. L.; Beljonne, D.; Coropceanu, V.; Cornil, J. Chem. Rev. 2004, 104, 4971. (29) Sánchez-Carrera, R. S.; Coropceanu, V.; Silva Filho, D. A.; Friedlein, R.; Osikowicz, W.; Murdey, R.; Suess, C.; Salaneck, W. R.; Bredás, J. L. J. Phys. Chem. B 2006, 110, 18904. (30) Zhang, W. W.; Zhu, W. J.; Liang, W. Z.; Zhao, Y.; Nelsen, S. F. J. Phys. Chem. B 2008, 112, 11079. (31) Warshel, A.; Parson, W. W. Q. Rev. Biophys. 2001, 34, 563. (32) Zhao, Y.; Liang, W. Z. Chem. Soc. Rev. 2012, 41, 1075.

ASSOCIATED CONTENT

S Supporting Information *

Functionals test and geometry properties of the diabatic states as well as the CDFT populations corresponding to the 3Bp−F state. This material is available free of charge via the Internet at http://pubs.acs.org. 12506



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(33) Voityuk, A. A. J. Phys. Chem. C 2010, 114, 20236. (34) Si, Y. B.; Zhong, X. X.; Zhang, W. W.; Zhao, Y. Chin. J. Chem. Phys. 2011, 24, 538. (35) Felts, A. K.; Pollard, W. T.; Friesner, R. A. J. Phys. Chem. 1995, 99, 2929. (36) Okada, A.; Chernyak, V.; Mukamel, S. J. Phys. Chem. A 1998, 102, 1241. (37) Davis, W. B.; Wasielewski, M. R.; Ratner, M. A.; Mujica, V.; Nitzan, A. J. Phys. Chem. A 1997, 101, 6158. (38) Segal, D.; Nitzan, A.; Davis, W. B.; Wasielewski, M. R.; Ratner, M. A. J. Phys. Chem. B 2000, 104, 3817. (39) Zhao, Y.; Liang, W. Z. J. Chem. Phys. 2009, 130, 034111. (40) Lin, S. H.; Chang, R.; Liang, K. K.; Shiu, Y. J.; Zhang, J. M.; Yang, T. S.; Hayashi, M.; Hsu, F. C. Adv. Chem. Phys. 2002, 121, 1. (41) Zhao, Y.; Liang, W. Z.; Nakumura, H. J. Phys. Chem. A 2006, 110, 8204. (42) Marcus, R. A. J. Phys. Chem. 1963, 67, 853. (43) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (44) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (45) Schwabe, T.; Grimme, S. Phys. Chem. Chem. Phys. 2007, 9, 3397. (46) Sousa, S. F.; Fernandes, P. A.; Ramos, M. J. J. Phys. Chem. A 2007, 111, 10439. (47) Cohen, A. J.; Sánchez, P. M.; Yang, W. T. Science 2008, 321, 792. (48) Zhao, Y.; Truhlar, D. G. Acc. Chem. Phys. 2008, 41, 157. (49) Chai, J. D.; Head-Gordon, M. Phys. Chem. Chem. Phys. 2008, 10, 6615. (50) Shao, Y.; Fusti-Molnar, L.; Jung, Y.; Kussmann, J.; Ochsenfeld, C.; Brown, S. T.; Gilbert, A. T. B.; Slipchenko, L. V.; Levchenko, S. V.; O’Neill, D. P.; et al. Phys. Chem. Phys. Chem. 2006, 8, 3172. (51) Yeganeh, S.; Voorhis, T. V. J. Phys. Chem. C 2010, 20756, 114. (52) Qin, H. M.; Zhong, X. X.; Si, Y. B.; Zhang, W. W.; Zhao, Y. J. Phys. Chem. A 2011, 115, 3116. (53) Lee, E.; Medvedev, E. S.; Stuchebrukhov, A. A. J. Chem. Phys. 2000, 112, 9015. (54) Hwang, H.; Rossky, P. J. J. Phys. Chem. B 2004, 108, 6723. (55) Mebel, A. M.; Chen, Y. T.; Lin, S. H. Chem. Phys. Lett. 1996, 258, 53. (56) Liang, W. Z.; Zhao, Y.; Sun, J.; Song, J.; Hu, S. L.; Yang, J. L. J. Phys. Chem. B 2006, 110, 9908. (57) Han, M. M.; Zhao, Y.; Liang, W. Z. J. Mol. Struct. (Theochem) 2007, 819, 13. (58) Zhang, W. W.; Liang, W. Z.; Zhao, Y. J. Chem. Phys. 2010, 133, 024501. (59) Kaduk, B.; Kowalczyk, T.; Voorhis, T. V. Chem. Rev. 2012, 112, 321. (60) Liang, K. K.; Chang, R.; Hayashi, M.; Lin, S. H. In Principles of Molecular Spectroscopy and Photochemistry; Chung-Hsin University Press: TaiChung, Taiwan, 2001. (61) Turro, N. In Modern molecular photochemistry; Univ. Science Books: Sausalito, Ca, 1991. (62) Wäckerle, G.; Bär, M.; Zimmermann, H.; Dinse, K. P.; Yamauchi, S.; Kashmar, R. J.; Pratt, D. W. J. Chem. Phys. 1982, 76, 2275. (63) Hochstrasser, R. M.; Scott, G. W.; Zewail, A. H. J. Chem. Phys. 1973, 58, 393. (64) Jankowiak, R.; Bässler, H. Chem. Phys. Lett. 1984, 108, 209. (65) Merrick, J. P.; Moran, D.; Radom, L. J. Phys. Chem. A 2007, 111, 11683. (66) Mijoule, C.; Leclercq, J. M. J. Chem. Phys. 1979, 70, 2560. (67) Chumakov, A. A.; Silvestrova, I. I.; Aleksandrov, K. Kristallogr. 1957, 2, 707. (68) Du, Y.; Ma, C.; Kwok, W. M.; Xue, J.; Phillips, D. L. J. Org. Chem. 2007, 72, 7148. (69) Fleischer, E. B.; Sung, N.; Hawkinson, S. J. Phys. Chem. 1968, 72, 4311. (70) Hoffman, R.; Swenson, J. R. J. Phys. Chem. 1970, 74, 425. (71) Alt, R.; Gould, I. R.; Staab, H. A.; Turro, N. J. J. Am. Chem. Soc. 1986, 108, 6911.

(72) Tahara, T.; Hamaguchi, H.; Tasumi, M. J. Phys. Chem. 1987, 91, 5875. (73) Kolev, T. M.; Stamboliyska, B. A. Spectrochim. Acta A 1999, 56, 119. (74) Gorman, A. A.; Rodgers, M. A. J. J. Am. Chem. Soc. 1986, 108, 5074. (75) Darmanyan, A. P.; Foote, C. S. J. Phys. Chem. 1993, 97, 4573. (76) Kowalczyk, T.; Wang, L. P.; Voorhis, T. V. J. Phys. Chem. B 2011, 115, 12135. (77) Fessenden, R. W.; Carton, P. M.; Shimamori, C. H.; Scaiano, J. C. J. Phys. Chem. 1982, 86, 3803. (78) Shimamori, H.; Uegaito, H.; Houdo, K. J. Phys. Chem. 1991, 95, 7664. (79) Place, I.; Farran, A.; Deshayes, K.; Piotrowiak, P. J. Am. Chem. Soc. 1998, 120, 12626. (80) Sigman, M. E.; Closs, G. L. J. Phys. Chem. 1991, 95, 5012. (81) Valiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; van Dam, H. J. J.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T. L.; et al. Comput. Phys. Commun. 2010, 181, 1477. (82) Becke, A. D. J. Chem. Phys. 1988, 88, 2547. (83) Farazdel, A.; Dupuis, M.; Clementi, E.; Aviram, A. J. Am. Chem. Soc. 1990, 112, 4206. (84) You, Z. Q.; Hsu, C. P.; Fleming, G. R. J. Chem. Phys. 2006, 124, 044506. (85) Olofsson, J.; Larsson, S. J. Phys. Chem. B 2001, 105, 10398. (86) Marchett, A. P.; Kearns, D. R. J. Am. Chem. Soc. 1967, 89, 768. (87) Ratner, M. A. J. Phys. Chem. 1990, 94, 4877.

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