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.
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.
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 .
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 ∼ η4 (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.
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
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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