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J. Phys. Chem. C 2010, 114, 20236–20239
Triplet Excitation Energy Transfer through Fluorene π Stack Alexander A. Voityuk* Institucio´ Catalana de Recerca i Estudis AVanc¸ats, Barcelona 08010, Spain, and Institut de Quı´mica Computational, Departament de Quı´mica, UniVersitat de Girona, 17071 Girona, Spain ReceiVed: September 3, 2010; ReVised Manuscript ReceiVed: October 21, 2010
Recently synthesized π-stacked polyfluorenes with cofacially oriented fluorene moieties are promising materials for molecular electronics. In the paper, I consider triplet excitation energy transfer (TEET) through the polyfluorene π stack. A key factor that controls the TEET rate is the electronic coupling of triplet excited states localized on neighboring moieties. Using Hartree-Fock configuration interaction singles calculations (HF/CIS-6-31G*), I study the conformational dependence of the TEET transfer integral and estimate the effective coupling between stacked fluorenes. To include the effects of conformational motion, the coupling is computed for 400 structures extracted from a molecular dynamics trajectory of the fluorene dimer. Structural fluctuations of the π stack are shown to cause large variations of the transfer integral. On the basis of the computed values of the TEET reorganization energy and the effective coupling, I estimate the absolute rate for hopping between neighboring fluorenes in the π stack. Conformational gating appears to play an important role in the energy transfer process. The triplet excitons in the π stack are found to be localized on single fluorene moieties. Introduction Triplet excitation energy transfer (TEET) is one of the key processes in molecular electronics,1-3 especially in organic lightemitting diodes (OLEDs). In OLEDs, recombination of electrons and holes generates excited triplet and singlet states in a ratio of 3:1. Thus, the rapid TEET over long distances is of primary importance for designing devices of high performance. Most computational studies on excitation energy transfer have been done for singlet excited states, and there is not much information on the mechanisms of TEET. Polyfluorene derivatives are among the most promising blueemitting materials.4 Usually, electronic communication between fluorene moieties occurs in molecular layers. Several years ago, Rathore et al. synthesized π-stacked polyfluorenes with cofacially oriented fluorene moieties.5 Effective overlap of π orbitals of fluorenes in the stack should provide effective electronic coupling between the subunits and facilitate electron and exciton transfer. Because of that, fluorene π stacks appear to be promising building blocks for the construction of organic onedimensional optoelectronic devices.6 Very recently, the TEET through π-stacked fluorene bridges has been studied using femtosecond transient absorption spectroscopy and quantum mechanical calculations.7 In particular, it has been shown that the dominant mechanism of TEET between donor and acceptor switches from tunneling to multistep hopping when the number of bridging fluorene units increases. Because experimental measurements of the absolute TEET rate between spectroscopically almost identical fluorene units are quite difficult,7,8 computational modeling can provide complementary data required for a detailed understanding of electronic and structural factors that control TEET.9,10 For instance, it allows one to explore how conformational dynamics affect the absolute rate of exciton transfer and to estimate the extent of excited-state delocalization within the system. As * To whom correspondence
[email protected].
should
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addressed.
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TEET probabilities between donor and acceptor significantly depend on the mutual arrangement of these sites,11,12 many thermally accessed structures have to be treated to obtain the effective coupling. This makes the development of theoretical models quite challenging. The TEET process can be presented as a simultaneous electron and hole transfer. When the initial and final states are localized on the donor and acceptor, the Marcus equation can be applied to estimate the hopping rate.7,13,14 The TEET rate is determined by three parameters: electronic coupling (V), driving force (∆G), and reorganization energy (λ)15
k)
(
1 -(∆G + λ)2 2π 2 V exp p 4λkBT √4πλkBT
)
(1)
The driving force ∆G is the energy difference of diabatic triplet excited states of the donor and acceptor. In the fluorene dimer, the donor and acceptor are identical; therefore, ∆G ) 0. The reorganization energy λ corresponds to a change in ∆G when moving the reactant to the product configuration without actually transferring the exciton. The dependence of the rate on conformational motion of the system is mainly controlled by the transfer integral V. Below, I report the results of a QM/MD study of TEET between adjacent fluorenes in a short π stack (Figure 1). Because triplet excited states of the dimer are found to be quite localized on single fluorene moieties, the hopping model can be applied. On the basis of configuration interaction singles (CIS-6-31G*) calculations of 400 structures of the π stack, I consider the effects of conformational dynamics on the triplet exciton hopping and estimate the absolute rate for this process. Computational Details Structure. To include the effects of conformational motion, MD simulation of the fluorene dimer (Figure 1) has been
10.1021/jp108407s 2010 American Chemical Society Published on Web 11/10/2010
Triplet Excitation Energy Transfer
J. Phys. Chem. C, Vol. 114, No. 47, 2010 20237
λ ) σ2(∆G)/(2kBT)
(4)
where kB is the Boltzmann constant and T ) 298 K. Results and Discussion
Figure 1. Structure of the fluorene dimer.
performed using the standard NVE ensemble protocol and the program AMBER.16 First, the structure of the system was optimized at the HF/6-31G* level with Gaussian 03 program,17 and the atomic charges were calculated using the RESP scheme.18 A 10 ns simulation was performed with the GAFF force field19 and 1 fs integration step. No bond length constraints were applied. Ten thousand snapshots separated by 1 ps were saved for subsequent structural analysis and quantum chemical calculations. Electronic Coupling. When two molecules interact in the excited state, the transition energies split. Within a two-state model, the electronic coupling for a symmetrical system may be estimated as a half of the excited-state splitting ∆.20 For nonsymmetrical systems, however, the half-splitting scheme provides only an upper limit of the coupling. Recently, Hsu et al.21 introduced the fragment excitation difference (FED) scheme to estimate the coupling for excitation energy transfer. The FED approach is an elegant extension of the fragment charge difference method22 used to calculate the electronic coupling for hole and excess electron transfer. Because electronic excitation can be viewed as the creation of an electron-hole pair, the excited-state electron density can be presented as a sum of the attachment and detachment electron densities.23 The TEET coupling is calculated by FED method21
|V| )
∆ · |xmn |
√(xmm - xnn)2 + 4xmn2
(2)
where ∆ ) |Em - En| is the energy difference of excited states m and n and the matrix elements xmn are expressed through the excited-state densities:
xmn )
∫r∈D Fexmn(r) dr - ∫r∈A Fexmn(r) dr
(3)
Fex is calculated as the sum of the attachment and detachment densities.21 The FED scheme can be used in combination with various quantum chemical methods. In the current study, I employ FED together with the Hartree-Fock CIS/6-31G* method. The calculations for 400 structures were performed using the program Gaussian03.17 The reorganization energy λ was derived from the variance σ of the TEET driving force ∆G (the energy difference of donor and acceptor diabatic states) along the MD trajectory24
π Stack Structure. The fluorene dimer (Figure 1) is a structural unit that carries information on the arrangement of the subunits in a fluorene π stack. Because of electrostatic repulsion, the symmetric C2V structure of the dimer has a higher energy than staggered conformations formed by right- and lefthand twisting of the system. Within 10 ns, 37 interconversions of the rotamers have been found (in average, the lifetime of each configuration is ca. 270 ps). The arrangement of fluorenes in the stack may be described by six parameters (three translations: shift, slide, and rise; and three rotations: tilt, roll, and twist; see Figure S1 in the Supporting Information). The right- and left-hand configurations differ by the sign of the slide and twist parameters. Mean values of the parameters (for the right-hand structure) obtained by processing the MD trajectory are as follows: shift ) 0.76 ( 0.30 Å, slide ) -0.91 ( 0.39 Å, rise ) 3.41 ( 0.18 Å, tilt ) 0.0 ( 2.9°, roll ) 17.1 ( 5.0°, and twist ) 24.1 ( 9.8°, which are in very good agreement with the parameters derived from single-crystal X-ray data5 of the dimer: shift ) 0.69 Å, slide ) -0.92 Å, rise ) 3.39 Å, tilt ) 0.4°, roll ) 20.5°, and twist ) 21.3. The variation of the stack parameters along the MD trajectory provides some details on structural dynamics of the system. In particular, the distance between fluorene planes, rise, does not change much (σ ) 0.18 Å), while shift and slide vary significantly. Because of limited flexibility of the methylene group, which connects the chromophores, the subunits are not exactly parallel to each other (the average angle between the planes is ca. 17°). A large variation of twist, ∼10°, should considerably affect the electronic coupling of the fluorene moieties. TEET Parameters. The CIS calculations of triplet excited states of dimer snapshots predict that the lowest excitation energy ranges within 2.6-2.9 eV in good agreement with the observed value of 2.8 eV.7 The extent, in which the lowest excited state is confined to a single fluorene moieties, depends on the dimer structure; on average, 90% of the exciton is localized on one of the chromophores. Because the subunits are equivalent (in average), the excited state is found equally often on the first and second subunits. According to experimental data,7 the triplet excitation energy of stacked fluorenes is almost independent of the number of subunits, indicating a lack of triplet delocalization. It should be noted, however, that the cooperative effects caused by the interaction of excited states in multichromophore systems are quite complicated.2,3,12 For instance, despite significant delocalization of singlet excited states in DNA π stacks, only a slight shift of their absorption with respect to the spectra noninteracting nucleobases is observed.25 In systems consisting of similar subunits, the extent of excited-state delocalization should depend on the ratio of the electronic coupling and the reorganization energy. For charge transfer in π stacks, Olofsson and Larsson found that the excess charge is delocalized, when 4V/λ > 1.26 This result seems to be also valid for TEET. According to my calculations, in the fluorene dimer 4V/λ ) 0.22 (V ) 0.02 eV and λ ) 0.37 eV, see below); therefore, the exciton should be localized on the monomers. The fluctuation of the TEET driving force in the stack is shown on Figure 2. Using eq 4, I have evaluated the reorganization energy for this process, λ ) 0.37 eV. When the donor and
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Figure 2. Fluctuation of the free energy ∆G of TEET in the fluorene stack.
Figure 3. Fluctuation of the TEET coupling of adjacent fluorenes.
acceptor electronic coupling is weak in comparison with the variation of the excitation energy gap (as found in the fluorene dimer), the dispersion of ∆ (with an appropriate sign) instead of the σ(∆G) can be used to estimate the reorganization energy. This approach gives λ ) 0.40 eV. Because the TEET coupling is very sensitive to conformational changes,9-12,21 its root mean square Vrms
Vrms ) √〈V2〉 )
1 n
n
∑ Vi2
(5)
i)1
should be used as a measure of the effective coupling. The dependence of the coupling is determined by variations of the energy gap ∆ and transition density matrix (see eq 2), due to structural fluctuations. Figure 3 demonstrates changes of the coupling squared within 400 ps. As seen, there are π stack conformations, where the TEET probability (proportional to V2) is by an order of magnitude larger than 〈V2〉. The alternating TEET active and inactive configurations are well separated. The calculations predict the effective coupling Vrms to be 2.01 × 10-2 eV. Interestingly, the measured triplet excitation energy of π-stacked fluorene bridges is independent of the number of subunits to within 0.02 eV.7 Figure 4 shows the contribution of TEET active conformations to the absolute rate (the structures are sorted by descending V2). As seen, 16% of all transitions is associated with 1% of the active configurations, and more than a half of TEET events are yielded by 5% of the substates. The analysis of the π stack conformations with strong electronic couplings suggests that in most cases slide ) 0.4-0.5 Å and |twist| < 10°. For instance, in snapshot 268 (the highest pick on Figure 3), shift, slide, and rise are 0.21, 0.42, and 3.45 Å, and tilt, roll, and twist are -4.9, 11.9, and -5.5°, respectively. I note that the symmetric structure, where all stack parameters (except for rise) are nearly zero, has the biggest coupling value. Because of its high strain energy, this configuration is, however, hardly accessible by thermal fluctuations. Most of the active π stack structures are found in
Figure 4. Conformational gating in the fluorene π stack. P is the portion of TEET events associated with the most active configurations. The structures are sorted by descending V2. As seen, 5% of conformations yield more than a half of TEET events.
relatively small region of the conformational space; therefore, the TEET process is conformationally gated.27 Hopping Rate. In the systems D-Fn-A, the states with a triplet exciton localized on any of the fluorene sites F are spectroscopically indistinguishable, and experimental measurements cannot provide the hopping rate for TEET between adjacent fluorenes.7 Using eq 1 and computed values of the effective coupling, V ) 0.020 eV, and the reorganization energy, λ ) 0.37 eV, I estimate the absolute rate for hopping between neighboring fluorenes to be 23 × 1010 s-1, and the corresponding hopping time is τ ) 4.4 ps. Thus, the TEET process in the π stack is very fast. This estimate is not completely quantitative, because of the approximation in the calculation of the reorganization energy. The calculations were performed for isolated system. Obviously, the interaction with environment will increase the reorganization energy and correspondingly decrease the TEET rate. The solvent reorganization energy for TEET is expected to be significantly smaller than that for electron transfer (∼1 eV). The TEET rate depends very sensitively on the reorganization energy; for a system with identical donor and acceptor, k ∼ exp(-λ/4kBT). If I take the solvent reorganization energy to be 0.3 eV and assume all other factors remain unchanged, the TEET rate would be 20 times lower, ∼1010 s-1. Note that the observed rates of TEET from benzophenone to the F2 bridge and from the bridge to naphthalene acceptor are also ∼1010 s-1.7 As already noted, the MD trajectory was generated for the electronic ground state of the fluorene π stack rather than for the corresponding excited state. Thus, the question arises as to whether the estimated TEET rate is considerably affected by sampling of the molecular geometries. The following argument suggests that this effect should be small. As demonstrated for electron transfer in π stacks, self-averaging of the coupling due to thermal structural fluctuations decreases considerably the extent to which the ET rate depends on the “observed” conformation of the system.28 Similarly, in spite of the TEET coupling between stacked fluorenes dramatically depends on their mutual arrangement, the effective coupling derived by averaging over thermally accessible configurations should be rather robust to changes in the stack structure. Furthermore, because the TEET process is not accompanied by considerable changes in the electrostatic interaction of the fluorene moieties, no significant differences in the π stack structure may be expected when the ground-state force field parameters rather than excited-state parameters (which are still unknown) are employed in the MD simulation.
Triplet Excitation Energy Transfer Conclusions In this study, I have calculated the effective electronic coupling and the reorganization energy for TEET between π-stacked fluorenes. To include the effects of conformational motion, MD simulation of the fluorene dimer in conjunction with CIS-6-31G* calculations was performed. The transfer integral is found to be very sensitive to conformational changes. The effective coupling value of 0.020 eV, which is derived by averaging over 400 snapshots, suggests an efficient electronic communication between stacked fluorenes. The π-π* triplet excited states in the system are found to be quite localized in agreement with experiment finding.7 The estimated TEET internal reorganization energy is 0.37 eV. The computed absolute rate of 23 × 1010 s-1 appears to be an upper limit for the TEET within fluorene π stacks. Conformational gating is shown to play an important role in the excitation energy transfer process. Acknowledgment. Support for this work comes from the Spanish Ministerio de Ciencia e Innovacio´n, Project No. CTQ2009-12346. Supporting Information Available: Figure of the parameters that define the arrangement of adjacent subunits in a π stack. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Bredas, J. L.; Beljonne, D.; Coropceanu, V.; Cornil, J. Chem. ReV. 2004, 104, 4971. (2) Scholes, G. D. ACS Nano 2008, 2, 523. (3) Scholes, G. D.; Rumbles, G. Nat. Mater. 2006, 5, 683. (4) Yu, W. L.; Pei, J.; Huang, W.; Heeger, A. J. AdV. Mater. 2000, 12, 828. (5) Rathore, R.; Abdelwahed, S. H.; Guzei, I. A. J. Am. Chem. Soc. 2003, 125, 8712. (6) Zhao, Y. S.; Fu, H.; Peng, A.; Ma, Y.; Liao, Q.; Yao, J. Acc. Chem. Res. 2010, 43, 409. (7) Vura-Weis, J.; Abdelwahed, S. H.; Shukla, R.; Rathore, R.; Ratner, M. A.; Wasielewski, M. R. Science 2010, 328, 1547. (8) Hayes, R. T.; Walsh, C. J.; Wasielewski, M. R. J. Phys. Chem. A 2004, 108, 3253.
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