Article pubs.acs.org/JPCA
Excitation Energy Transfer in Donor-Bridge-Acceptor Systems: A Combined Quantum-Mechanical/Classical Analysis of the Role of the Bridge and the Solvent Stefano Caprasecca* and Benedetta Mennucci* Dipartimento di Chimica e Chimica Industriale, University of Pisa, Via Risorgimento 35, 56126 Pisa, Italy ABSTRACT: The technical application of excitation energy transfer requires a fine control of the geometry of the system. This can be achieved by introducing a chemical bridge between the donor and acceptor moieties that can be tuned in its chemical properties and its length. In such donor-bridge-acceptor systems, however, the role of the bridge in enhancing or depleting the energy transfer efficiency is not easy to predict. Here we propose a computational strategy based on the combination of time-dependent density functional theory, polarizable molecular mechanics and continuum approaches. The resulting three-layer model when applied to the study of the energy transfer process in different porphyrin-based systems, each characterized by a specific donor/acceptor pair and various types of bridges, allows us to dissect the role of through-bond and through-space mechanisms and clarify their dependence on the nature and length of the bridge as well as on the presence of a solvent.
1. INTRODUCTION
where VDA is the electronic coupling between donor (D) and acceptor (A) moieties and JDA is the spectral overlap integral between the donor emission and the acceptor absorption. As shown in eq 1, the rate depends on the spectroscopic properties of the D/A pair (through JDA) as well as on their interaction through VDA. As a matter of fact, it is the coupling VDA, the quantity that can be tuned to obtain the best performances, once the D/A pair has been selected. The coupling, in fact, can be largely changed by modifying the geometry of the D/A assembly through molecular linkers or by playing with the properties of the external environment. The dependence of the coupling on the geometrical characteristics of the system is generally modeled by assuming a dipole−dipole interaction, eventually screened by the solvent, as originally suggested by Förster
Excitation energy transfer (EET) is a ubiquitous process present in natural systems to efficiently collect energy from sunlight. The same process can also be exploited in artificial devices for technical applications. In that case, however, the efficiency has to be accompanied by reproducibility and predictability; both of these characteristics require a fine control of the geometry of the EET system, which can be achieved if the transferring moieties are held together by interactions strong enough to keep their relative orientation and distance fixed. These interactions can be either intermolecular (often of stacking type) or intramolecular in terms of chemical bonds. In the latter case, the tuning and the control of the EET process will be made much easier by playing on the nature and the length of the molecular bridge linking the transferring moieties. In this context, it is evident that an accurate prediction of how the bridge mediates the interactions leading to the EET process is of fundamental importance in the design of optimal devices. The modeling of the EET process has undergone a very important development in the past few years because of the application of quantum-mechanical approaches, possibly combined with classical formulations to include the effects of the environment. In the general case, a weak coupling hypothesis is used; in that case, the EET rate can be defined, as suggested by Förster several decades ago1 starting from the Fermi golden rule; namely, it is possible to write kEET =
2π |VDA|2 JDA ℏ
VDA ≈
(2)
where μ⃗ TD and μ⃗ TA are the D and A transition dipole moments, respectively, R⃗ DA is the D/A distance vector, and κ is a geometrical factor depending on the mutual orientation of R⃗ DA, μ⃗TD, and μ⃗ TA. The solvent screening is defined as the inverse of the square of the refractive index n of the solvent. One of the most recognizable features of the dipole−dipole approximation is the R−6 dependence of the EET rate on the D/A distance, which can be employed experimentally to Special Issue: Franco Gianturco Festschrift Received: March 21, 2014 Revised: April 25, 2014 Published: April 25, 2014
(1) © 2014 American Chemical Society
T T 1 μD μA κ 3 n2 RDA
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compute inter- and intramolecular distances.2 However, there are several cases when such dependence is not valid, one of the most relevant being the case when the dipole−dipole approximation no longer holds (at short D/A distances, when RDA is smaller than the dimension of the chromophores, or for elongated systems3−5). This issue can be tackled by employing models overcoming the dipole−dipole approximation, such as the transition density cube6 or other approaches where the actual shape of the D/A transition densities is used.7,8 In addition, the environment effects can also introduce a further dependence on both the distance and the orientation of the two interacting moieties, as it has been shown by combining the QM description of the coupling in terms of full transition densities with a classical but polarizable description of the environment either using a continuum or a molecular mechanics description.9,10 Moreover, at short distances, another issue must be taken into account: the overlap of the chromophores’ orbitals leads to the onset of a purely quantum mechanical effect, that is, an electronic exchange interaction, and the type of energy transfer no longer follows Förster model but decays exponentially with the distance, according to Dexter model.11 Apart from these effects, the presence of a bridge (B) between D and A can also induce deviations from the Förster behavior, even at large D/A separations.12 These deviations have most frequently been ascribed to superexchange,13−15 a longer-range exchange mechanism allowed by the bridge (or also other intervening structures between the donor and acceptor moieties). However, another possible explanation of the deviations observed in bridge-mediated singlet−singlet EET rates is that the bridge polarizability can explicitly affect the Coulomb interaction between the D/A moieties.16−19 The study and analysis of such deviations in porphyrinic systems with a rigid bridge is the aim of this paper. In covalently linked porphyrin arrays, a semirigid linker is generally introduced between the donor and acceptor chromophores (metal- and free-base-porphyrins, respectively) to keep a well-defined and rigid structure and maintain some properties of the isolated chromophores so that the resulting complex has predictable characteristics while imparting efficient electronic communication channels among the chromophores.20,21 It is well known that the bridge-mediated EET mechanism in these systems involves both through-space (TS) and through-bond (TB) contributions. Recently, we have studied ethyne-linked zinc- and free-base-tetraarylporphyrin dimers showing that the EET process is tuned by the type of linker and by substitution on the porphyrin rings. Using a TDDFT description combined with a recently developed fully polarizable quantum-mechanical/discrete/continuum method,22 we have dissected the bridge-mediated contributions to the energy transfer in terms of TB interactions and Coulomb (TS) terms mediated by the polarizability of the bridge. We found that bridge-mediated superexchange contributions largely boost energy transfer between the porphyrin units, and only by including these effects together with those of the solvent can a good agreement with the experiments be achieved.22,23 Here we further extend the analysis by comparing two sets of porphyrinbased systems, each characterized by a specific donor/acceptor pair and various types of bridges. By analyzing the role of bridge-mediated through-bond and through-space mechanisms and their dependence on the nature and length of the bridge, we can get a useful insight into the nature of the energy-transfer
process in this type of systems as well as on the ways it is affected by the presence of the environment.
2. METHODS AND COMPUTATIONAL DETAILS The computational method employed has been described elsewhere22 and will be only quickly reviewed here. It combines a continuum dielectric description of the solvent with a flexible definition of the regions that will be described through quantum mechanics or in terms of a polarizable molecular mechanics (MMPol) approach. For what concerns the continuum approach, the integral equation formalism (IEF)24 version of the polarizable continuum model (PCM)25 is used, whereas in the polarizable MM description fixed charges and induced dipoles are employed (MMPol10). More in detail, the electronic coupling between donor and acceptor units is calculated from transition densities obtained for the two units, separately. Once these densities, ρT, are known, the coupling can be written as V0 =
∫ dr′ ∫ dr ρDT*(r′) |r −1 r′| ρAT (r) ∫ dr′ ∫ dr ρDT*(r′)gxc(r′, r)ρAT (r) − ω0 ∫ dr ρDT *(r )ρAT (r ) +
(3)
where gxc is the exchange-correlation kernel determined by the specific functional used, whereas the last term is an overlap contribution weighted by the resonance transition energy ω0. In eq 3, the first term represents the Coulomb interaction between the transition densities and, in the present systems, is the largely dominant term, with the other two contributions being almost negligible. This Coulomb term can be seen as the generalization of the point-dipole/point-dipole coupling, where the transition dipole moments appear instead of the transition densities. The classical environment implies the inclusion of two additional terms, VPCM and VMMPol, describing the electrostatic interaction between the acceptor transition density with the PCM charges and MMPol dipoles, respectively, induced by the excitation in the donor Vsolv =
⎡
∑ ⎢∫ dr ρAT (r) t
⎣
⎡
−
1 ⎤ ⎥q (st ; ϵ(ω), ρDT ) |r − st | ⎦ t
∑ ⎢∫ dr ρAT (r) p
⎢⎣
rp − r ⎤ ⎥μ (r p ; ρ T ) D |r − rp|3 ⎥⎦ p
(4)
where the frequency-dependent dielectric permittivity ϵ(ω) reduces to the optical permittivity, ϵ∞ (the square of the refractive index), if a nonequilibrium response for the environment is used. On top of this, the presence of the continuum solvent and of the MM charges and dipoles also affects the D and A transition densities (and all of the related transition properties), thus implicitly changing the Coulomb, exchange -correlation, and overlap terms of the coupling with respect to the value calculated for the same system in the gas phase. This is automatically accounted for by solving the timedependent density functional theory (TD-DFT) equations in the presence of solvent- (PCM) and bridge- (MMPol) induced terms. To assess the influence of the bridge on the coupling, we have built three models:23 (a) model M0, a full-QM approach 6485
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into point charges available in Gaussian09 asking for the Gassian03 defaults.
which completely neglects the bridge; (b) model Mc, a full-QM approach, which includes the bridge at QM level; and (c) model MMPol, a hybrid QM/MM where the bridge is described using a polarizable MM approach. (See Figure 1.)
3. RESULTS To investigate the mechanisms behind the EET process in D/ B/A systems, here we compare two sets of porphyrin-based systems, each characterized by a specific type of donor and acceptor pair (indicated as A and B pair) and various types of bridges; see Figure 2. These systems have been deeply analyzed
Figure 1. (a) Schematic representation of the three models employed. Colored red and blue lines indicate fragments treated at QM level of theory, which are assigned to either the donor (red) or the acceptor (blue) subsystem. Black dots indicate classical sites that are assigned a charge and a polarizability and which are treated following the MMPol method. Model M0 completely disregards the bridge; model MMPol includes it at a classical MMPol level; model Mc assigns part of the bridge to the donor moiety and part to the acceptor one, always cutting through the triple bond. (b) Definition of the two chromophore−bridge angles φ and ψ.
Figure 2. Geometry of the two sets of systems studied, A and B. These differ by the substituents on the porphyrin rings, while the bridges are the same.
by Albinsson and coworkers32 (series A) and by Lindsey and coworkers20,33 (series B), from both an experimental and a modellistic point of view. In all systems here studied, energy transfer occurs from the photoexcited zinc-porphyrin (the donor) to the free-baseporphyrin (the acceptor). The excitations involved are the (weak) Q bands, localized on the two porphyrins, which are virtually degenerate for the Zn-porphyrin and slightly split for the free one. 3.1. Through-Space Mechanism. We start the study from the A set of systems, which have been deeply investigated by Albinsson and coworkers32 using absorption and steady-state and time-resolved emission spectroscopy. These systems correspond to the nB series of Figure 2. As a first analysis, in Figure 3, we report the dependence of the EET rate with respect the D/A distance for the gas-phase systems using the three models (M0, Mc, and MMPol) described in the previous section. Note that instead of reporting the rate constants we focus on the sum of the squared couplings for each pair of excitations involving the two quasidegenerate Q states (first and second excited states for both D 2 2 and A moieties), that is, V2 = V21→1 + V1→2 + V22→1 + V2→2 . (The indices refer to the excitation states of the donor and the acceptor, respectively.) If we assume the Förster definition of the rate reported in eq 1 to be valid and assume that the spectroscopic overlap J does not depend on the bridge, the two analyses are exactly equivalent. Both of these approximations are confirmed by the experimental observation that absorption and emission spectra of dimeric system are practically equivalent to those of the single units.32 Here and in the
We note that in models M0 and MMPol the QM part is identical and only contains the porphyrinic rings with the related substituents, whereas in model Mc the partition into donor and acceptor can be done in various ways; for each bridge, we have tried to associate half to D and half to A, always cutting through the triple bond, as shown in Figure 1. In a previous study, we have, however, shown that the results do not change if the bridge is partitioned otherwise.23 All QM calculations were run at the TD-DFT level, employing the CAM-B3LYP26 functional and the 6-31G(d) basis set, using a locally modified version of the Gaussian09 suite of codes.27 The MMPol part of the system was described using fixed MM charges obtained from a fit of the electrostatic potential of the molecule or fragment according to the Merz and Kollman method28,29 at the CAM-B3LYP/6-31G(d) level, plus a set of polarizable sites (coincident with the MM atoms) described by isotropic polarizabilities. We employed the Thole model,30 which avoids intramolecular overpolarization problems by using a smeared dipole−dipole interaction tensor. Atomic isotropic polarizability values were taken from the fit of experimental molecular polarizabilities performed by van Duijnen and Swart.31 The presence of covalent bonds between the QM and the MM fragments was tackled using hydrogens to saturate the free valences. The IEFPCM calculations have been done using the cavity and the surface discretization procedure 6486
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models, respectively); we also note that this behavior does not change when the solvent is included, because in all cases the slope is about −7. (2) The MMPol and Mc models return remarkably similar results at all bridge lengths; from this we conclude that the effects of the bridge can be properly described at a classical level and that superexchange effects are not significant. However, by comparing M0 and MMpol results we also observe an important effect of the bridge that, if neglected, would lead to a large overestimation of the rate. (3) The bridge screening factors reported in Table 1 are always smaller than 1 (and they decrease in solution). Therefore, the bridge actually screens the D/A interaction. Note also how this screening remains approximately constant within the nB series. (4) The inclusion of the solvent through the PCM significantly reduces the D/A interaction, as shown by effective screening factors around 0.63. This value is larger than what is predicted by Förster, being 1/n2 = 0.49 for this solvent. Note also that s̃sol is not constant, but it decreases slightly along each nB series as the bridge length increases. This analysis on the EET rate of A systems, showing an almost pure TS character, is consistent with the experimental findings, confirming the almost R−6 dependence with respect to the bridge length, but it completely reverts to the conclusions drawn in the original paper,32 which reported a large underestimation of the rate when a TS description was used. To better understand this apparent contradiction, we need to better detail the TS estimate done by Albinsson and coworkers, which is based on the Förster expression of the rate
Figure 3. Calculated squared couplings versus D/A distance for A systems in gas-phase using the Mc, MMPol, and M0 models (logarithmic scale on both axes). Calculated slopes are −6.9 (Mc), −7.1 (MMPol), −7.1 (M0).
following analyses the D/A distance RDA is calculated as the distance between the Zn atom and the center of the free porphyrin ring. To quantify the effects of the bridge on the EET rate, we have defined an intrinsic bridge screening, which is both model(M,sol) and solvent-dependent, namely, s bdg ≡ (V 2(M,sol)/ V2(M0,sol)). As in previous papers,9,34 we can also define a solvent screening as the ratio between the total coupling and the Coulomb term only, both calculated in solution: ssol ≡ Vtot/ VCoul. This effective screening can be directly compared with the 1/n2 factor appearing on the dipole−dipole approximation of the coupling reported in eq 2. In the present analysis, however, it is more useful to redefine the effective screening factor as s̃sol ≡ (V2tot/V2Coul)1/2 where the squared couplings are obtained as the sum of the four different contributions between the first two excited states of D and A. The results obtained for the bridge and solvent screening in gas-phase and in dichloromethane (DCM) solution are shown in Table 1. From results reported in Figure 3 and in Table 1, we can point out four main findings: (1) As shown in Figure 3 for the vacuum case, the squared coupling dependence reduces with the D/A distance according to the relation V2 = cRn, with n around −7 (the slopes in the plot are −6.9, −7.1, and −7.1 for the Mc, MMPol, and M0
kForster = ̈
bridge
RDA
s(MMpol) bdg
s(Mc) bdg
s̃sol
vacuum
2B 3B 4B 5B 6B 2B 3B 4B 5B 6B
20.0 26.9 33.8 40.7 47.5 20.0 26.9 33.8 40.7 47.5
0.70 0.69 0.70 0.70 0.70 0.59 0.55 0.55 0.55 0.55
0.66 0.63 0.67 0.69 0.73 0.62 0.56 0.57 0.56 0.57
0.66 0.65 0.64 0.63 0.62
DCM
a
(5)
where ΦD and τD are the quantum yield and the lifetime of the donor (both defined in absence of the acceptor), NA is the Avogadro number, n is the refractive index of the solvent, and RDA is the distance between donor and acceptor. This expression is a direct consequence of assuming a dipole− dipole interaction to describe the coupling (see eq 2). In their analysis, ΦD and τD are obtained experimentally by a spectroscopic study on a D system, which also includes the bridge fragment (as in our Mc model), whereas RDA is set equal to the distance between the centers of the two porphyrinic rings and the orientation factor, κ, to an average 5/6 value. To have a more direct analysis of the Förster approximation, we have calculated the transition dipoles obtained for both donor and acceptor units and estimated the square of the coupling, assuming the same RDA value used in the reference paper. This analysis has been repeated for both the M0 and Mc models. The results obtained show that the ratio V2/V2dip varies from 0.31 (2B) to 0.67 (6B) for the Mc model and from 0.29 (2B) to 0.75 (6B) for the M0 model. The M0 data clearly show the inaccuracy of the assumption that the transitions can be described as dipolar: in that model, actually, no bridge is present so each moiety cannot be perturbed by the bridge in any way. However, the dipole−dipole estimate of the rate is significantly smaller than the corresponding rate obtained considering the real transition densities. The picture does not change with the Mc model in which both donor and acceptor transition dipoles have been obtained in the presence of the bridge.
Table 1. Bridge and Solvent Effects Calculated for A Systems in Vacuo and in Dichloromethane (DCM) Using the MMPol and Mc Modelsa solvent
2 9000 ln 10 ΦDκ J 6 4 128π 5NA τDRDA n
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In conclusion, the description given in ref 32 of the EET in these systems as dominated by through bridge effects seems to be significantly biased by the assumption that the TS contribution can be simulated with a dipole−dipole interaction. Moreover, their estimate of the solvent effects on the rate through a constant factor equal to (1/n2)2 is too large, and this further reduces the predicted TS contribution. If we now move to the B set of systems (originally named ZnFbU by Lindsey and coworkers20,33), the analysis changes completely, as it will be shown in the next section. 3.2. Through-Bond Mechanism. Apparently the B and the A systems considered above are very similar: the bridge is identical, and changes are limited to the porphyrin substituents. However, it has been demonstrated in several occasions (see, for instance, Strachan et al.33 and, recently, Caprasecca et al.23) that the porphyrin substituents in such systems may markedly affect both the efficiency and the nature of the electronic energy transfer. The data calculated for the B set of systems, reported in Table 2, clearly point out these differences. In particular, three main specificities are found:
Figure 4. B dimer in gas-phase and solvents: V2MMPol versus RDA (both axes in logarithmic scale). The fit curves, appearing as straight lines, are of the form: V2MMPol = c1R−α. The fitting parameters are reported in Table 3. The calculations have been carried out in vacuo, dichloromethane (DCM), methanol (MeOH), and benzene.
Table 2. TS Contribution and Solvent Effects Calculated for B Systems in Vacuo and in Dichloromethane (DCM)a
Table 3. Fitting Parameters Obtained in the Different Solvents Characterized by the Refractive index n
solvent
bridge
RDA
vacuum
2B 3B 4B 5B 6B 2B 3B 4B 5B 6B
19.6 26.5 33.3 40.2 47.1 19.6 26.5 33.3 40.2 47.1
DCM
a
s̃sol
χTS%
0.78 0.66 0.56 0.50 0.47
9.93 2.85 1.18 0.71 0.53 5.87 1.07 0.38 0.23 0.20
V2MMPol = cR−α
V2med = Ae−βR
solvent
n
α
β
vacuum DCM MeOH benzene
1.00 1.43 1.33 1.50
9.0 10.8 10.1 12.2
0.18 0.22 0.21 0.23
Because the model used, MMPol, does not include any QM contribution mediated by the bridge (e.g., superexchange), we expect that this discrepancy can be attributed to a classical, polarization-driven contribution that also couples to the solvent effects. In previous works32 a so-called “mediation rate constant” was defined as the difference between the full rate and the TS contribution, generally estimated through the Förster eq 5, to investigate the superexchange energy transfer rate dependence on the bridge length. Moreover, the electronic coupling for superexchange is generally assumed to approximately decay exponentially with the distance. Translating this analysis within our modellistic framework, we can define a mediation squared coupling V2med as
Distances are in angstroms.
(1) Contrary to what was observed for the A systems, here the couplings calculated with the Mc model are at least one order of magnitude larger than those calculated with the MMPol one. As a result, the TS contribution to the coupling calculated from the relation χTS = V2MMPol/V2Mc is always
