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# Dexter energy transfer pathways

Edited by Jay R. Winkler, California Institute of Technology, Pasadena, CA, and accepted by Editorial Board Member Harry B. Gray May 16, 2016 (received for review August 31, 2015)

## Significance

Controlling the dynamics of excitons—including their transport, fission, fusion, and free carrier generation—presents a central challenge in energy science, optoelectronics, and photobiology. We develop a coupling-pathway theory for triplet energy transfer, a process controlled by the structure of the medium between donor and acceptor sites, and find two competing coupling pathway mechanisms. At shorter distances or high tunneling gaps, the electron and hole move sequentially from donor to acceptor, accessing donor–acceptor charge-transfer exciton virtual states; at longer distances or lower tunneling gaps, virtual exciton states of the bridge mediate the transport. Molecular design strategies can leverage these competing mechanisms and their distinctive dependences on molecular structure.

## Abstract

Energy transfer with an associated spin change of the donor and acceptor, Dexter energy transfer, is critically important in solar energy harvesting assemblies, damage protection schemes of photobiology, and organometallic opto-electronic materials. Dexter transfer between chemically linked donors and acceptors is bridge mediated, presenting an enticing analogy with bridge-mediated electron and hole transfer. However, Dexter coupling pathways must convey both an electron and a hole from donor to acceptor, and this adds considerable richness to the mediation process. We dissect the bridge-mediated Dexter coupling mechanisms and formulate a theory for triplet energy transfer coupling pathways. Virtual donor–acceptor charge-transfer exciton intermediates dominate at shorter distances or higher tunneling energy gaps, whereas virtual intermediates with an electron and a hole both on the bridge (virtual bridge excitons) dominate for longer distances or lower energy gaps. The effects of virtual bridge excitons were neglected in earlier treatments. The two-particle pathway framework developed here shows how Dexter energy-transfer rates depend on donor, bridge, and acceptor energetics, as well as on orbital symmetry and quantum interference among pathways.

- Dexter energy transfer
- triplet excitons
- triplet energy transfer
- two-particle coupling pathways
- superexchange

A compelling challenge in supramolecular chemistry is to direct the flow, fission, and fusion of excitons in molecular assemblies (1⇓⇓–4). When donor or acceptor species undergo a spin change during energy transfer, a two-particle or Dexter interaction enables the energy transfer because the Förster (dipole–dipole) coupling is spin forbidden (5). Developing design principles for Dexter energy transfer is a considerable challenge compared with that of single-electron (hole) transfer because of the combinatorial growth in the number of mediating (virtual) two-particle states with system size (6⇓⇓–9). As with single-particle (electron or hole) transfer, Dexter energy transfer arises from donor–acceptor coupling mediated by molecular species (10). Here, we develop a coupling pathway theory for bridge-mediated Dexter energy transfer and explore the relative contributions of bridge and donor–acceptor charge-transfer excitons to the transport.

A wide variety of critical chemical systems rely on bridge-mediated Dexter transfer of triplet excitons. The lowest-energy electronic excited states of transition metal complexes used for solar-energy harvesting are often high spin, and the excitation energy usually flows to a low-spin ground state acceptor (3). In the electro-optics underpinning light-emitting diodes based on metal-containing chromophores, the exchange of energy between low- and high-spin excited states is crucial for device efficiency (11). As well, protection of biological light-harvesting machinery from damage induced by sensitized singlet oxygen formation relies on a Dexter energy transfer quenching mechanism (12). The strong dependence of the Dexter coupling on the bridge structure indicates that triplet energy-transfer materials offer additional control (compared with the case for Förster energy transfer) through the manipulation of the bridge-mediated coupling.

Dexter’s 1953 analysis of spin-forbidden excitation energy transfer between donor (D) and acceptor (A) moieties in contact invoked coupling via the electron–electron Coulomb operator (5). However, most Dexter systems of interest today involve chemically bridged species. In addition to the two-electron interaction identified by Dexter, one-electron interactions (applied to second or higher order) also couple D to A. The term “Dexter coupling” is now understood to arise from both one- and two-electron interactions that may be mediated by a bridge (see *Two-State EnT Kinetics*), and two-state approximations to the Dexter coupling that include both contributions are well known (13). Pioneering kinetic studies of bridge-mediated Dexter energy transport in molecules have been reported by Closs et al. (14), Albinsson et al. (15), Harriman et al. (16), and Spieser (10); and considerable recent attention has turned to Dexter energy transfer at nanoparticle–molecule junctions (4). Despite the crucial role played by bridge-mediated Dexter energy transfer, a general framework to assess coupling pathway-mediated Dexter interactions and their interferences is lacking. We formulate a theory for bridge-mediated Dexter coupling pathways that allows the appraisal of specific coupling mechanisms.

Our description of Dexter coupling pathways relies on a configuration-interaction single-excitations (CIS) framework, motivated by schemes used to assess bridge-mediated interactions for single-electron/hole transfer (6, 7), adapted here to track the coupled motion of two particles. Pathway decompositions allow molecular-level understanding of energy, orbital symmetry, and interference effects on energy-transfer rates. The framework developed here allows analysis of Dexter-pathway coupling mechanisms in the language of virtual exciton pathways mediated by the bridge. We find that Dexter pathways through short bridges with high tunneling-energy gaps are dominated by charge-transfer virtual exciton intermediates [donor–acceptor charge-transfer excitons (DAE)] with one particle (electron or hole) on D and the other on A. The coupling in this short-distance high-barrier regime is consistent with an early conjecture of Closs et al. (14) and with the picture of Harcourt et al. (13). At longer distances or lower bridge energy gaps, however, bridge-localized virtual excitons (without DAE intermediates) dominate the Dexter coupling. These virtual excited states of the bridge, or bridge excitons (BE), are characterized by electron-hole pairs localized on the bridge. We provide formulas to assess the BE contribution to the Dexter coupling, because the earlier theories did not account for these BE intermediates.

We denote the donor, bridge, and acceptor chemical fragments in the energy transfer (EnT) system as D, B, and A, respectively. To describe the electron/hole charge distributions in these regions we use a *B*, or *A*, where the plus sign indicates a hole, and the minus sign indicates an electron). For an exciton with electron and hole localized in separate regions *R* and *i* and excited electron in orbital *x* is denoted

## Two-State EnT Kinetics

Nonadiabatic triplet-to-triplet (tr) EnT is well described in the golden-rule approximation when the (resonant) donor and acceptor electronic transitions are at much lower energies than all other electronic transitions. The golden rule rate is

A commonly used expression for the bridge-mediated Dexter coupling is given in Eq. **2** (19). We find that Eq. **2** does not capture these crucial BE contributions to the Dexter coupling, and we provide more general formulas that account for the BE contributions. The approximate Dexter coupling between D-centered

In Eq. **2**, **2** indicates that

In Eq. **2**,

The GF approach is useful to interpret **2** describes the contribution of single-particle transfer (SPT) pathways to the triplet-EnT coupling **2** suggests that this contribution always involves DAE virtual intermediates with charge distributions **2** excludes an important class of triplet BE virtual mediating states

## Characteristics of the Dexter Coupling

Experimental and theoretical studies of Dexter transport have been carried out in rigid and flexible molecules, in polymers, in polymer assemblies, and in metal–organic frameworks (3, 24, 25). Dexter rates drop approximately exponentially with distance (10), and Eq. **2** suggests a distance decay constant equal to the sum of the electron and hole superexchange decay constants (14). Experimental studies of Harriman found that some Dexter rates decay with exponential decay constants as small as 0.1 Å^{−1} for Ru(II)–Os(II) terpyridyl complexes linked by 1,4-diethynylene-2,5-dialkoxy-benzene bridges (16). Albinsson et al. found exponential decay constants of 0.45 Å^{−1} for phenylene ethynelene linked porphyrins (15). For alkane linkers, Closs et al. found large decay exponents, 2.8 Å^{−1} (14). Computed decay constants as large as 3.4–3.8 Å^{−1} were reported by Curutchet and Voityuk for through-solvent Dexter transport (26). Experimental and theoretical studies clearly indicate that Dexter couplings depend on the structure and energetics of the bridge.

## CIS Model in a Localized Basis

We use a CIS approach (27) to describe the tr Dexter coupling. CIS methods were found to describe tr EnT couplings accurately in earlier studies (21⇓–23). We use an orthogonal basis of natural localized molecular orbitals (NLMOs) that are mostly two-center bonding (e.g., *σ* and *π*) and two-center antibonding orbitals (e.g., *i* *x*

The NLMO representation for *i* and *x* produces an intuitive interpretation of a triplet basis state

The Hamiltonian elements among CIS basis states are (29)*i* (*x*); each off-diagonal Fock matrix element *i* and orbital *j* (orbital *x* and orbital *y*) (28, 29)**3**) into diagonal **5**) contains the electron and hole NLMO orbital energies (**6**) contains one-particle

The term BE used for the

## Exact EnT Splittings in Model Compounds

We focus on a simple set of *n*-alkyl–bridged dienes and norbornanes (Fig. 2) to study bridge-mediated EnT couplings, including their distance, energy-gap, molecular-conformation, and coupling-pathway dependence. For all of the molecules in Fig. 2 we choose the donor and acceptor segments to be the left (L) and right (R) C=C bonds, and we set **3**) given by *V*_{tr} between **3**:**3**) with all elements containing the

## Contributions of DAE and BE Virtual Intermediates to the Dexter Coupling

In our quantum computations, we used restricted Hartree–Fock methods implemented in Gaussian 09 (30) with a 6–31G basis. Fig. 3*A* shows *A*. Fig. 3*A*, *Inset* indicates the relative magnitude of the DAE and BE contributions [*A* and 3*A* show that the BE contribution in extended alkane bridges with more than seven to eight CC bonds is larger than the DAE contribution. The relative BE contribution is larger for bridges with smaller tunneling barriers. To explore this switching effect, we shift the energies of all bridge NLMO diagonal Fock matrix elements, **5**, so that the energy gaps **3**) equally delocalized over *B* shows *A*, where we have set the *A*]. For the seven-bond bridge in Fig. 3*B*, the lowest BE eigenstate *A*). Therefore, for the lower barrier systems in Fig. 3*B*, the BE contribution dominates the coupling for all bridge lengths, becoming more than two orders of magnitude larger than the DAE contribution for longer bridges. Fig. 3*B* shows that the BE contribution produces large

To investigate the effects of molecular conformations on the alkane systems, we sampled structures by choosing random torsional angles and optimizing these conformations with restricted Hartree–Fock methods using a 6–31G basis set (RHF/6–31G). The folded structures thus generated (Fig. 2*B*) were used to compute *B*). In most cases, the BE contribution is greater than or approximately equal to the DAE contribution. The Dexter couplings for the conformationally sampled alkane bridges are smaller compared with the couplings for the extended alkane bridges (for the seven-CC bond bridge, *A*, and for the partially folded seven-CC bond bridges in Table S1,

The trends in the coupling mechanism apply to more complex bridged structures. Tables S3 and S4 show *C*) where, as with the linear alkanes, we choose the donor and acceptor segments to be the L and R C=C bonds, and we set *C*). In Table S4, we use the same structures as in Table S3 with lowered energy gaps. The *D*). In this structure, the Dexter coupling is symmetry forbidden (

To summarize, the splitting computations find that the Dexter coupling is mediated by BE virtual states, rather than by DAE virtual states; i.e.,

## Triplet-EnT Pathways

Having established the importance of BE contributions to the Dexter coupling **2**. Our focus is the first single-particle transfer (SPT) term, which is a product of D-to-A electron transfer (ET) and hole transfer (HT) couplings. To understand the contributions of this term to **3**) in the *Supporting Information*). We also define **S6** and **S7**). That is, we use equations identical to Eq. **8** where we zero out all

## Donor—Acceptor Exciton vs. Bridge-Exciton Triplet-EnT Pathways

We derive a generalized GF expression for the first (SPT) term of Eq. **2**. This term contains electron-transfer **8**, we replace the total CIS Hamiltonian **3**) with a Hamiltonian **6** are ignored (ne means no exchange). Therefore, **5**) [containing the Coulomb attraction terms **8**), we obtain pathway expressions for the SPT components of the total Dexter coupling, of the DAE-mediated coupling, and of the BE-mediated coupling. These expressions (Eqs. **S15**, **S19**, and **S21**) are denoted

where **10** describes DAE pathways and is given by**S25**). **10** describes BE pathways**S26**). *K* of virtual intermediate states shown in Fig. 1. **9**.

The DAE contribution **11** is the generalized GF pathway expression for the first (SPT) component of Eq. **2**. It describes EnT as a sequence of two complete D-to-A electron and hole tunneling steps (first term, **11** contains all of the upper and lower tunneling paths connecting

The bridge exciton contribution, **12**), describes all tunneling pathways from

## Rapid Growth in the Number of Bridge-Exciton Intermediate States with Chain Length

Ignoring pure exchange when computing the Dexter coupling is not generally sound. For the systems studied in Fig. 2*A*, the average exchange contribution to the Dexter coupling in the long chain limit (Fig. 2*A*) is about 25% of **2**) without keeping the *N* bonding/antibonding orbitals (**12** for *N* grows, the number of possible BE virtual intermediates **11**). Thus, BE pathways are important for long bridge lengths or low energy gaps, where omitting the BE contribution to the Dexter coupling may introduce errors of one to two orders of magnitude.

## Conclusions

We have found that bridge-exciton tunneling pathways dominate triplet energy transfer mediation in the long-distance/small tunneling gap regime accessed in many molecular structures of current interest. As well, we have developed a coupling pathway description for bridge-mediated triplet Dexter coupling. The Dexter coupling is exponentially sensitive to donor–acceptor distance and to bridge structure, suggesting that these EnT rates and their directionality may be manipulated by the bridge structure. As with bridge-mediated electron and hole transfer, control can be realized by using pathway interference effects, bridge energetics, and through-bond/through-space coupling trade-offs. The theory enables an atomic-level description for the origins of Dexter coupling, a necessary step toward controlling Dexter coupling interactions in a wide range of systems of current interest in energy science and molecular biophysics.

The most significant result of the Dexter pathway analysis is the demonstration that virtual bridge-exciton intermediate states (Fig. 1, center) can dominate the EnT coupling for long bridges and low tunneling-energy bridges. This BE-mediated coupling, and thus the Dexter coupling, cannot be expressed as a simple product of electron and hole donor-to-acceptor tunneling steps. Indeed, Curutchet and Voityuk’s studies of Dexter couplings through solvent found Dexter decay exponents to be smaller than the sum of the electron- and hole-mediated superexchange coupling decay exponents (26). The coupling pathway dissections introduced here are sufficiently general to enable the further development of structure–function relations for Dexter energy-transfer interactions.

## Pathway Analysis

The full CIS Hamiltonian

To proceed with the Löwdin projection method we write the identity operator in the

From the above definitions, and using the projection technique, we can obtain a GF expression (**S5** can reproduce the value of

The Löwdin projection expression **S5** where

To derive single particle pathway contributions to the EnT coupling, we define a Hamiltonian operator without pure exchange **5** of main text). The matrix elements of the off-diagonal Hamiltonian

Replacing **S5** gives**S12**,**S8**].

In summary, the GF expression

From Eq. **S15**, we can obtain an exact expression for the effective EnT coupling mediated by the DAE (and not the BE) denoted **S16**. Namely,**S16**; i.e.,

To simplify the expression for **S16** as**S15** gives

The terms

For the compounds under study and for all energy gaps mentioned in the main text (Figs. 2 and 3), we find that

Tables S1–S4 are Dexter coupling computations on the folded alkane systems (Fig. 2*B*) and the polynorbornyl bridged systems Fig. 2*C*).

## Acknowledgments

We thank Joe Subotnik, Marshall Newton, and Greg Scholes for fruitful discussions. S.S.S. thanks the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013), under the Research Executive Agency Grant 609305. D.N.B., C.L., and A.M.V. thank the National Science Foundation (Grant DMR-1413257) and Duke University for support. S.S.S. and D.N.B. also thank the Institute of Advanced Studies at the University of Freiburg, Germany for support of this collaboration.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: david.beratan{at}duke.edu or skourtis{at}ucy.ac.cy.

Author contributions: S.S.S., C.L., P.A., A.M.V., and D.N.B. designed research; S.S.S., C.L., P.A., and A.M.V. performed research; S.S.S., C.L., P.A., A.M.V., and D.N.B. analyzed data; and S.S.S., C.L., P.A., A.M.V., and D.N.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. J.R.W. is a Guest Editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1517189113/-/DCSupplemental.

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- Abstract
- Two-State EnT Kinetics
- Characteristics of the Dexter Coupling
- CIS Model in a Localized Basis
- Exact EnT Splittings in Model Compounds
- Contributions of DAE and BE Virtual Intermediates to the Dexter Coupling
- Triplet-EnT Pathways
- Donor—Acceptor Exciton vs. Bridge-Exciton Triplet-EnT Pathways
- Rapid Growth in the Number of Bridge-Exciton Intermediate States with Chain Length
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