Dissecting the Nature of Exciton Interactions in Ethyne-Linked Tetraarylporphyrin Arrays

We investigate how electronic energy transfer in a series of three ethyne-linked zinc- and free base tetraarylporphyrin dimers is tuned by the type of linker and by substitution on the porphyrin rings. We use time-dependent density functional theory (TD-DFT) combined with a recently developed fully polarizable QM/MM/PCM method. This allows us to dissect the bridge-mediated contributions to energy transfer in terms of superexchange (through-bond) interactions and Coulomb (through-space) terms mediated by the polarizability of the bridge. We explore the effects of the substituents and of the bridge-chromophore mutual orientation on these contributions. We find that bridge-mediated superexchange contributions largely boost energy transfer between the porphyrin units. When the effect of the solvent is also considered through the polarizable continuum model (PCM), we find good agreement with the through-bond versus through-space contributions determined experimentally, thus indicating the need to properly incl...


Introduction
Multiporphyrin architectures have been largely investigated as potential efficient photonic devices; indeed they possess many attractive features: a rigid and planar geometry, a high stability, an intense electronic absorption and a small HOMO-LUMO gap; moreover, their optical and redox properties can be tuned by appropriate metallation. [1][2][3] Different types of covalently linked porphyrin arrays with linear, cyclic, and cross-linked geometries have been constructed and used to achieve a thorough understanding of all factors that affect electronic communication among the various constituents. 4,5 Actually, only through a detailed analysis of these factors and their coupling it is possible to develop a rational design of a wide range of molecular devices to be used for photonic applications.
An important part of the investigations reported so far on multiporphyrin architectures has been focussed on electronic energy transfer (EET) processes, which together with electron transfers determine the electronic communication. In these arrays a major issue concerns the extent of communication that is mediated by the linker. In covalently linked porphyrin arrays, indeed, a semirigid linker is generally introduced between the donor and acceptor chromophores (metal-and free base-porphyrins, respectively) in order 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. 6,7 Among the linkers having these characteristics, ethyne linkage between aryl groups have been used in combination with different porphyrin constituents, because of their small attenuation factors. 8 In particular, Lindsey and collaborators have conducted detailed spectroscopic investigations, both static and time-dependent, to rationalize the role of different factors in determining the EET efficiency, and more in general the electronic communication in complexes where the porphyrin constituents are linked via diphenylethyne groups at the meso positions (see Figure 1 for a representation of the complexes investigated). 6 Figure 1: Schematic representation of the geometry of the three porphyrin dimers studied.
From their analysis on different Zinc-and free base-porphyrin dimers they concluded that energy transfer primarily proceeds via a process mediated by the diarylethyne linker but, remarkably, they also observed that the rate of such transfers can change by orders of magnitude if the chromophores are slightly modified, despite keeping the same linker. For example the energy transfer rates of two diphenylene ethynylene-linked bisporphyrin systems (F 30 ZnFbU and ZnFbU, see Figure 1) differ by a factor 10 despite the fact that they contain the same linker. As the energy transfer in ZnFbU is extremely rapid and essentially quantitative, the slower dynamics observed for F 30 ZnFbU was attributed to attenuated energy transfer and not to the onset of competing processes associated with the fluorination of the phenyl rings. The main issue is therefore to understand the mechanism beyond the different EET efficiencies due to minor changes in the chromophores and the real role played by the linker.
It is well known that the EET mechanism in these systems involves both through-space (TS) and through-bond (TB) contributions. The former is mediated by the standard Coulomb interaction between the two transition densities localized on the donor and acceptor moieties, while the TB contribution involves interactions beyond the coulombic terms and is generally explained in terms of D/A orbital overlap, i.e. an electronic exchange interaction. Historically the TS mechanism was modelled using the Förster dipole-dipole approximation 10 (see Equation 3), while a Dexter-like picture was introduced for the TB one, where the EET rate constant presents an exponential decay with the D/A distance. 11 Deviations from what is expected on the basis of the two idealized models have most frequently been ascribed to superexchange, 12-14 a longer-range exchange mechanism allowed by the presence of intervening structures, such as linkers, between the donor and acceptor moieties. Another possible explanation of the deviations sometimes observed in bridge-mediated singlet-singlet EET rates is that the bridge polarizability can explicitly affect the Coulomb interaction between the D/A moieties. 15 4 ), in order to dissect the contributions to the electronic communications. In particular, the aim is to quantify the TS vs. TB contribution, and also to give a detailed explanation of the role played by the linker and by different substituents on the D/A moieties in the final EET efficiency. Note that in the previous sentence, and in the rest of the paper from here, we will use a definition of TS and TB contributions consistent to that employed by Lindsay and collaborators, i.e., we will label TB the bridge-mediated contribution to the total coupling that is QM in character, mediated by orbital overlap and exchange, and TS the Coulomb contribution, which can be described including the bridge classical polarizability. Such definitions differ from the standard ones in that they both include the effects of the bridge.

Methods and computational details
The method we use to analyse the EET properties of the porphyrin dimers combines a continuum dielectric description of the solvent with either a fully quantum mechanical (QM) description of the porphyrin dimer, or alternatively a QM description of the photoactive moieties combined with a polarizable MM description of the linker.
The continuum solvent description is based on the IEF formulation of the PCM model, 22 using the discretization from surface charge density into point charges available in Gaussian09 (requiring the Gaussian03 defaults). Within this framework, the QM system is embedded in a cavity of shape and dimension defined according to the geometrical structure of the solute, and thus also depending on the distance and relative orientation of the chromophores. The solvent is described as a polarizable continuum (characterized by its dielectric constant and refractive index), which responds to the presence of the QM system through a set of induced (or apparent) charges placed on the surface of the molecular cavity. In turn, such charges act back on the QM system from which they are generated: this mutual polarization effect is solved through a modified self-consistent field scheme. In addition, within the PCM framework it is possible to introduce nonequilibrium effects that arise whenever a fast process in the QM system originates delays in the response of the solvent.
This is exactly what happens during an electronic excitation or during an EET, when the energy donor, D, is de-excited by exciting the acceptor, A. In the common time-scales of these processes, the response of the solvent is incomplete in the sense that only its fast degrees of freedom (of electronic nature) can equilibrate with the final state of the QM system, while the rest remains frozen in the initial configuration corresponding to the QM system before the change.
On the other hand, the MMPol method adopts an atomistic description of the environment based on a classical polarizable force field based on the induce dipole model. 21 where g xc 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 ω 0 .
Within the combined MMPol/PCM framework, 19 the polarization effect of the solvent and the linker on the electronic coupling is not introduced as a screening factor as in the Förster model but it enters in the definition of the coupling itself through an additional term (called explicit term) which sums to the V 0 defined in Equation 1. Particularly, if we mimic the solvent and linker polarization induced by the donor transition density in terms of PCM charges and MM induced dipoles, respectively, such an additional term becomes: where the frequency-dependent dielectric permittivity ε(ω) reduces to the dynamic or optical permittivity, ε ∞ (the square of the refractive index), if a nonequilibrium response for the environment is used. The presence of the solvent and the MM charges and dipoles of the bridge also affects the coupling in an implicit way, by affecting the D and A transition energies and densities and all the related transition properties. This is automatically accounted for by solving the LR equations, which now include explicit solvent-(PCM) and bridge-(MMPol) induced terms.
All the QM calculations were run at the TD-DFT level, employing the CAM-B3LYP functional and the 6-31G(d) basis set, using a locally modified version of the Gaussian09 suite of codes. 25 The MMPol 21 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 method 26,27 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, 28 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. 29 The presence of covalent bonds between the QM and the MM fragments was tackled by following the link atom method: 30 the QM-MM bonds were initially cut and saturated on both sides with hydrogens. The saturated MM fragment, isolated, was then used to obtain the Merz-Kollman charges; afterwards, the MM atoms previously bound to the QM chromophores, together with their saturation hydrogens, were removed to avoid hyperpolarization problems; their MM charges were summed and distributed onto the covalently bound MM atoms.

Results
In all the systems here studied, energy transfer occurs from the photoexcited zinc-porphyrin (Zn, the donor) to the free base-porphyrin (Fb, 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 Fb one. The comparison between the experimental absorption spectra of the isolated porphyrins and that of the ZnFb complex shows no marked modification, either in shape or in position, thus indicating a relatively weak electronic interaction. 9 The spectra of the fluorinecontaining Zn porphyrins resemble those of their non-fluorinated counterparts, while the oscillator strength of the S 0 → Q-state transition in fluorinated Fb is reduced by 40% with respect to the non-fluorinated one.

Excitation energies
As a preliminary analysis we have compared experimental absorption maxima for the different porphyrins with the calculated vertical energies. All calculations have been performed on ground-state optimized structures and they include the effects of the toluene solvent through a PCM description.

Energy transfer
According to the experimental analysis carried out by Strachan et al., 9 the EET mechanism remains predominantly through-bond (TB) rather than through-space (TS) in all the systems, even if the rate is considerably slower in F 30 ZnFbU than in all the other arylethyne-linked ZnFb dimers. The observed energy transfer rate was therefore assumed to be due to the additive effects of the TB (k TB ) and TS (k TS ) processes, where the (approximate) Förster formula was used to calculate the TS transfer rate: Here, κ is an orientation factor, Φ D and τ D the fluorescence quantum yield and lifetime of the donor chromophore in the absence of the acceptor, I the Förster spectral overlap term (in mmol −1 cm 6 ), n the solvent refractive index, and R DA the donor-acceptor centre-to-centre distance (in cm).
The TB contribution was derived from the relations: χ TB + χ TS = 1 (5) and the results obtained in toluene are reported in Table 2.
As it can be seen from the Table, the TS energy transfer rates for all three dimers are relatively but the bridge is here described using the polarizable MM approach presented above (see Figure   2). We note that in the  Table 3 in terms of the sum of the squared couplings for each pair of excitations involving the two quasi-degenerate Q states (first and second excited states for both D and A moieties), i.e., (the indices refer to the excitation states of the donor and the acceptor, respectively). For the systems with two conformers ('P' and 'T'), a Boltzmann average has been computed in order to obtain effective couplings. For this purpose, the free energy of the monomers has been calculated in vacuo, so the weights in vacuo and in toluene are assumed to be the same (this is indeed a reasonable approximation due to the very low polarity of toluene). We first analyse the results relative to the Mc model, for all dimers, both in vacuo and in toluene.
Concerning the effect of the conformation, we note that it only slightly affects the couplings.
Moreover, we note that the calculated couplings for the ZnFbU dimer are quite larger than those for the other two systems. In particular, the squared couplings calculated for the fluorinated dimer  Table 2, where the ZnFbU system shows the fastest experimental EET rate (41.7 ns −1 ), compared to those of ZnFbB(CH 3 ) 4 (8.7 ns −1 ) and F 30 ZnFbU (4.2 ns −1 ), the latter being one order of magnitude slower than its nonfluorinated counterpart. Below, we will try to understand why and how the presence of the F atoms affects so markedly the electronic coupling and therefore the EET rate.
For what concerns the effect of the solvent, this is homogeneous in the three systems and it always leads to a significant reduction of the coupling, due to a dominant screening effect: the reduction is ∼30% for both fluorinated and non fluorinated ZnFBU and ∼20% for ZnFbB(CH 3 ) 4 .
The smaller screening in the ZnFbB(CH 3 ) 4 system is probably due to the fact that the lateral groups in the linker make the PCM cavity larger than in the other two systems; as a result, the solvent effect is reduced and so is the screening.
We now focus on the results obtained from the three different models Mc, M0 and MMPol. From the data reported in Table 3, comparing M0 and MMPol results, we note that the fluori- It is interesting to observe that the largest contribution from the linker is found in the fluorinated

Effect of fluorination
In order to clarify whether the differences between the ZnFbU and F 30 Table 3, are remarkably similar to those of ZnFbU. We therefore conclude that the effect of the F atoms does not lie in the distortion of the nuclear geometry, but rather in the perturbation of the electron density.
In Table 4 we report the transition dipoles of the donor and acceptor chromophores, and the orientation factor κ, for some systems, calculated in vacuo. The factor κ is helpful to understand how the mutual orientation of these dipoles affects the coupling, in a Förster-like, dipole-dipole interaction picture. The Förster coupling in vacuo is calculated as: where µ T D and µ T A are the donor and acceptor transition dipoles, respectively, and R DA the donoracceptor vector distance; the terms in square brackets correspond to the definition of the orientation factor κ. The data reported in Table 4, although quantitatively approximate since relying on a dipoledipole approximation, is however useful to show that the effect of the fluorination is dual; on one hand it causes a reduction of the transition dipole moments: see for instance µ T D,1 and µ T A,1 , that are reduced by nearly 3-and 2-fold, respectively, when passing from non-fluorinated systems to F 30 ZnFbU. At the same time, the absolute value of the orientation factors involving the first state of the donor (κ 1,1 and κ 1,2 ) are also markedly reduced.
Even in such a simplistic picture, the combination of the two effects explains fairly well the observed reduction of the squared couplings. The remarkably consistent results of the two nonfluorinated systems ZnFbU and H 30 ZnFbU, despite their different structure, point out that such a reduction can be explained in terms of the perturbation of the chromophore electron densities due to the presence of the F atoms.

Through-space and through-bond contributions
Finally, we move to the comparison with the experimental data reported in Table 2. To do that we define the TS contribution to the total coupling, relative to a method M, as the fraction of the squared coupling obtained with that method, relative to the squared coupling obtained with the most complete Mc method, namely: We have computed the TS contribution for the M0 and MMPol models, both in vacuo and in toluene. The comparison with the results by Strachan et al. 9 for the three porphyrin dimers under study are reported in Table 5. For ZnFbU and F 30 ZnFbU, the Boltzmann average of the results relative to the two conformers is shown. The data in Table 5  introducing the solvent effects, χ TS significantly changes for all the systems. These changes are different both in module and in sign for the three systems: we observe a net increase of the TS contribution for F 30 ZnFbU, from 5 to 16 %, bringing our results closer to the 18 % contribution derived experimentally. On the other hand, the TS contribution decreases markedly for the ZnFbU system, and even more so for the ZnFbB(CH 3 ) 4 dimer. For these also we note that the solvent-induced changes lead towards a much better agreement with the Förster estimates from experimental data, but while the agreement in absolute terms is fairly good for the ZnFbU and F 30 ZnFbU dimers, our calculated TS contribution is twice the Förster one in the case of the ZnFbB(CH 3 ) 4 dimer.
In order to better understand the origin of such differences in the results for the different systems, we have further investigated the possible role played by torsional motions of the planes of the two porphyrins with respect to that of the linker.

Twisting motion
Several geometries of the ZnFbB(CH 3 ) 4  We first analyse the results for the ZnFbB(CH 3 ) 4 dimer, shown in Table 6. Note that, because of   , calculated in toluene, is shown in Figure 5. From the analysis of the two systems, we can say that, statistically, the torsional motion around the equilibrium geometry value reduces the TS contribution for ZnFbB(CH 3 ) 4 , while it leaves the one calculated for ZnFbU virtually unaltered. In both cases, the calculated results tend to agree with the experimental observation.

Conclusion
We have studied three ZnFb porphyrin dimers linked by a semirigid phenyleneethynylene spacer, focusing on the effects that different linkers and substituents have on the electronic coupling, and comparing our results, when possible, to the experimental values by Strachan et al.. 9 We have employed our mixed QM/discrete/continuum method, where the QM parts of the system are described at the TD-DFT level, the discrete ones following the MMPol classical polarizable description, and the solvent is introduced as a structureless continuum. Employing models which differentiates by the way the linker is accounted for, has allowed us to estimate the through-bond and through-space contributions to the total coupling.
Our results are in general good agreement with the experimental ones, and in particular we verify the the fluorination strongly affects the electronic communication among the dimer moieties, despite the fact that the system geometry is mostly unaltered. We also obtain values for the TS contribution to the total coupling that are in agreement with the experimental ones, particularly for the two systems ZnFbU and F 30 ZnFbU, characterized by the same linker. For the third system, ZnFbB(CH 3 ) 4 , our results on the TB contribution seem to indicate that the twisting motion of the porphyrin planes with respect to the bridge plane may play a more relevant role than one would expect, particularly if we consider that the methyl groups on the linker make this structure more rigid than the others.
In general, our approach performs fairly well in describing the porphyrin dimers examined. As predicted in several experimental and theoretical studies on similar systems, the highly conjugated linker greatly affects the electronic communication between the energy donor and acceptor, mostly through a superexchange mechanism involving the bridge orbitals. This, on one hand, poses a serious limit to a QM/discrete description of the system, since the bridge polarization has only a marginal effect on the total coupling. Our MMPol model, therefore, cannot be expected to provide a correct description of the energy transfer process, as it did with other systems studied elsewhere (see for instance the PDI-TDI dimer in Curutchet et al. 15 and Caprasecca et al. 19 ). On the other hand, when the results from the MMPol model are compared to those given by the full-QM model, Mc, we are able to estimate the TB and TS contributions and evaluate how these are affected by, for instance, structural modifications or solvation. Finally, it is important to stress that this last term, solvation, introduced through the PCM method, again proves essential to correctly describe real systems and obtain reasonable results.