^{1,a)}, Mark S. Hybertsen

^{2,b)}and David R. Reichman

^{1,c)}

### Abstract

We apply our theoretical formalism for singlet exciton fission, introduced in the previous paper [T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, J. Chem. Phys.138, 114102 (Year: 2013)]10.1063/1.4794425 to molecular dimers of pentacene, a widely studied material that exhibits singlet fission in the crystal phase. We address a longstanding theoretical issue, namely whether singlet fission proceeds via two sequential electron transfer steps mediated by charge-transfer states or via a direct two-electron transfer process. We find evidence for a superexchange mediated mechanism, whereby the fission process proceeds through virtual charge-transfer states which may be very high in energy. In particular, this mechanism predicts efficient singlet fission on the sub-picosecond timescale, in reasonable agreement with experiment. We investigate the role played by molecular vibrations in mediating relaxation and decoherence, finding that different physically reasonable forms for the bath relaxation function give similar results. We also examine the competing direct coupling mechanism and find it to yield fission rates slower in comparison with the superexchange mechanism for the dimer. We discuss implications for crystalline pentacene, including the limitations of the dimer model.

After this work was submitted, we became aware of the work of Havenith *et al.*,^{55} which, as we acknowledged above, provides a picture in qualitative agreement with that presented here. We thank Josef Michl for bringing this work to our attention, as well as for sharing a draft of Ref. 59 before publication. We also thank Eran Rabani for the use of his two-electron integral FFT code. This work was supported in part by the Center for Re-Defining Photovoltaic Efficiency through Molecule Scale Control, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0001085. This work was carried out in part at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Contract No. DE-AC02-98CH10886 (M.S.H). T.C.B. was supported in part by the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under Contract No. DE-AC05-06OR23100.

I. INTRODUCTION

II. METHODOLOGY

A. Geometry and electronic structure

B. Quantum dynamics

III. RESULTS FOR PENTACENE

A. Scanning energies

B. Superexchange and the strength of the electronic coupling

C. The effect of the bath

D. Direct mechanism

E. Covalently linked dimer

IV. CONCLUSIONS

### Key Topics

- Charge transfer
- 30.0
- Superexchange interactions
- 25.0
- Phonons
- 21.0
- Excitons
- 15.0
- Energy transfer
- 12.0

## Figures

Schematic representation of the five electronic states relevant for singlet fission in a dimer. The actual states employed in the calculations are spin-adapted linear combinations yielding overall spin-singlets, unlike those shown here.

Schematic representation of the five electronic states relevant for singlet fission in a dimer. The actual states employed in the calculations are spin-adapted linear combinations yielding overall spin-singlets, unlike those shown here.

Molecular geometry of the pentacene crystal. Three pentacene molecules are emphasized, displaying the three symmetry-unique nearest-neighbor dimer pairs discussed in the text. Also shown are isosurface plots of the HF HOMO (a) and LUMO (b) of the isolated molecules, including the phase convention adopted in this work.

Molecular geometry of the pentacene crystal. Three pentacene molecules are emphasized, displaying the three symmetry-unique nearest-neighbor dimer pairs discussed in the text. Also shown are isosurface plots of the HF HOMO (a) and LUMO (b) of the isolated molecules, including the phase convention adopted in this work.

Singlet fission yield, *P* _{ TT }(*t*) × 200%, after the four periods of time indicated for the [1/2 1/2] pentacene dimer. The dashed line qualitatively separates the superexchange (SX) regime, *E*(*CT*) > *E*(*S* _{1}), from the sequential (SEQ) regime, *E*(*S* _{1}) > *E*(*CT*). Estimated energy levels for the pentacene dimer are denoted by the white circle.

Singlet fission yield, *P* _{ TT }(*t*) × 200%, after the four periods of time indicated for the [1/2 1/2] pentacene dimer. The dashed line qualitatively separates the superexchange (SX) regime, *E*(*CT*) > *E*(*S* _{1}), from the sequential (SEQ) regime, *E*(*S* _{1}) > *E*(*CT*). Estimated energy levels for the pentacene dimer are denoted by the white circle.

Energy level diagram depicting the diabatic electronic states (i.e., before mixing) and the excitonic electronic states (i.e., after mixing), for a typical “superexchange” energy configuration indicative of a pentacene dimer. For exciton states which are a significant mixture of two different types of diabatic states, the notation *i* ↔ *j* is employed.

Energy level diagram depicting the diabatic electronic states (i.e., before mixing) and the excitonic electronic states (i.e., after mixing), for a typical “superexchange” energy configuration indicative of a pentacene dimer. For exciton states which are a significant mixture of two different types of diabatic states, the notation *i* ↔ *j* is employed.

Population dynamics contrasting superexchange and sequential CT-mediated singlet fission, shown in both the diabatic and exciton bases. Diabatic energy levels for panels (a) and (c) are *E*(*S* _{1}) − *E*(*TT*) = 250 meV, *E*(*CT*) − *E*(*TT*) = 500 meV; and for panels (b) and (d) are reversed, i.e., *E*(*S* _{1}) − *E*(*TT*) = 500 meV, *E*(*CT*) − *E*(*TT*) = 250 meV.

Population dynamics contrasting superexchange and sequential CT-mediated singlet fission, shown in both the diabatic and exciton bases. Diabatic energy levels for panels (a) and (c) are *E*(*S* _{1}) − *E*(*TT*) = 250 meV, *E*(*CT*) − *E*(*TT*) = 500 meV; and for panels (b) and (d) are reversed, i.e., *E*(*S* _{1}) − *E*(*TT*) = 500 meV, *E*(*CT*) − *E*(*TT*) = 250 meV.

The same as in Fig. 3 but for population dynamics calculated by the NIBA-type master equation, which is perturbative to second order in the electronic couplings, *V* _{ ij }.

The same as in Fig. 3 but for population dynamics calculated by the NIBA-type master equation, which is perturbative to second order in the electronic couplings, *V* _{ ij }.

Dramatic slowing down of singlet fission dynamics for decreasing electronic coupling strength η (a); note that the time axis is in log-scale. The numerically extracted fission rate obeys the predicted superexchange scaling *k* ∼ η^{4} (b), however the equilibrium population of *TT* decreases with increasing coupling, due to enhanced mixing with non-*TT* states (c).

Dramatic slowing down of singlet fission dynamics for decreasing electronic coupling strength η (a); note that the time axis is in log-scale. The numerically extracted fission rate obeys the predicted superexchange scaling *k* ∼ η^{4} (b), however the equilibrium population of *TT* decreases with increasing coupling, due to enhanced mixing with non-*TT* states (c).

Calculated fission rate for a pentacene dimer with varying system-bath coupling, quantified by the reorganization energy, λ. Secular, Markovian Redfield theory (filled circles) predicts a linear dependence, which is known to be accurate for small λ but becoming more inaccurate for large λ (indicated by the shaded region). Realistic values for pentacene are λ ≈ 50–150 meV, which reliably predicts a fission rate *k* ≈ 2–10 ps^{−1}, i.e., τ ≈ 100–500 fs, in reasonably good agreement with experimental rates of 80–200 fs.

Calculated fission rate for a pentacene dimer with varying system-bath coupling, quantified by the reorganization energy, λ. Secular, Markovian Redfield theory (filled circles) predicts a linear dependence, which is known to be accurate for small λ but becoming more inaccurate for large λ (indicated by the shaded region). Realistic values for pentacene are λ ≈ 50–150 meV, which reliably predicts a fission rate *k* ≈ 2–10 ps^{−1}, i.e., τ ≈ 100–500 fs, in reasonably good agreement with experimental rates of 80–200 fs.

Three different forms of the spectral density investigated here, along with the electronic eigenvalue differences for pentacene (orange vertical sticks). The overlap between these energy differences and the spectral density, i.e., the ability to absorb and emit resonant phonons, largely determines the rate of population transfer and hence singlet fission.

Three different forms of the spectral density investigated here, along with the electronic eigenvalue differences for pentacene (orange vertical sticks). The overlap between these energy differences and the spectral density, i.e., the ability to absorb and emit resonant phonons, largely determines the rate of population transfer and hence singlet fission.

Singlet fission population dynamics for the three spectral densities depicted in Fig. 9 .

Singlet fission population dynamics in the absence of *CT* states, for varying values of the direct electronic coupling element given in the legend. Sub-picosecond fission is only observed for the unphysically large value of 20 meV, to be contrasted with theoretical estimates ranging from 5 to less than 1 meV. ^{ 28,55,59–61 }

Singlet fission population dynamics in the absence of *CT* states, for varying values of the direct electronic coupling element given in the legend. Sub-picosecond fission is only observed for the unphysically large value of 20 meV, to be contrasted with theoretical estimates ranging from 5 to less than 1 meV. ^{ 28,55,59–61 }

Two different views of a covalently linked pentacene dimer akin to the tetracene dimers of Refs. ^{ 24 } and ^{ 25 } . The electronic couplings are estimated to be significantly smaller than in the crystal phase dimer pairs investigated above, predicting a much smaller rate of fission.

Two different views of a covalently linked pentacene dimer akin to the tetracene dimers of Refs. ^{ 24 } and ^{ 25 } . The electronic couplings are estimated to be significantly smaller than in the crystal phase dimer pairs investigated above, predicting a much smaller rate of fission.

## Tables

Electronic coupling parameters (in meV) of pentacene for the three dimer types described in the text. Values in parentheses are those calculated by Yamagata *et al.* ^{ 40 } and Troisi and Orlandi, ^{ 41 } the latter only where available (*t* _{ HH }). The perfect discrepancy in sign for the [1/2 1/2] and [−1/2 1/2] dimers suggests a difference in the adopted phase convention between our work and that of Ref. ^{ 40 } .

Electronic coupling parameters (in meV) of pentacene for the three dimer types described in the text. Values in parentheses are those calculated by Yamagata *et al.* ^{ 40 } and Troisi and Orlandi, ^{ 41 } the latter only where available (*t* _{ HH }). The perfect discrepancy in sign for the [1/2 1/2] and [−1/2 1/2] dimers suggests a difference in the adopted phase convention between our work and that of Ref. ^{ 40 } .

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