A Marcus-Hush perspective on adiabatic singlet fission

Singlet fission (SF) is a process which occurs in a pair of chromophores whereby one singlet excited state separates into a pair of triplet states.¹ In solid materials and solution, the triplets become independent and could potentially be harvested in solar energy conversion devices. As such, it is possible to generate more than one electron and hole from an incoming photon. An ideal solar cell harnessing singlet fission is limited to an efficiency of 45.9% under the AM1.5G solar spectrum,^2,3 substantially higher than the Shockley-Queisser limit (33.7%), which applies to single junction solar cells.⁴ It is hoped that actual SF-solar cells based on an underlying crystalline silicon cell will exceed 30% efficiency.^5–8 

Substantial interest in tetracene as a singlet fission material is motivated by its energetic compatibility with silicon.^5,7,9 The T₁ state lies at 1.25 eV,¹⁰ just above the crystalline silicon band gap of 1.12 eV. Furthermore, singlet fission in tetracene is endothermic, a feature of theoretical SF solar cells at the thermodynamic efficiency limit.³ The endothermic nature of SF in tetracene would seem to contradict the apparent temperature independence of SF in evaporated thin films.^11–13 However, as shown by Piland and Bardeen, upon annealing, an Arrhenius dependence in the SF rate can be restored.¹⁴ Accelerated, temperature-independent SF in microcrystalline films is thus attributed to cofacial defects. The cofacial defects promote excimer formation which is observed at low temperature but may also enhance coupling between the optically bright singlet and dark triplet-pair potential energy surfaces.

The role of the excimer state in singlet fission has been disputed. Its observation as an intermediate has been claimed in solutions of TIPS-pentacene, which demonstrates exothermic SF.¹⁵ Also, an excimer intermediate with triplet-pair spectroscopic character has been claimed in TIPS-tetracene solutions.¹⁶ However, as shown by Dover et al. by comparison with sensitized triplet-triplet annihilation, in TIPS-tetracene solutions, the excimer serves as a trap.¹⁷ This is also the case in covalent tetracene dimers.¹⁸ In peropyrene, the excimer state was held responsible for negligible singlet fission yields, despite promising energetics in solution.¹⁹ Similar reasoning has explained the relative SF yields in isobenzofuran derivatives.²⁰ In perfluoropentacene, triplet pair formation is observed before excimer formation.²¹ The interplay between excitonic coupling and accelerated singlet fission is of great interest.²² 

To explore the interplay between the excited singlet, the triplet pair, and the excimer state, one can turn to ab initio quantum chemistry with a suitable methodology agnostic to the level of electron excitation.^18,23–26 Quantum chemical calculations have greatly improved our understanding of the singlet fission process, and an overview is well captured by the review of Casanova.²⁷ While many investigations approach singlet fission from the diabatic standpoint, the natural view from an ab initio perspective is under the auspices of the adiabatic Born-Oppenheimer approximation.

It would be desirable from a conceptual perspective, to view singlet fission (and triplet-triplet annihilation, TTA) in terms of the interplay between adiabatic potential energy surfaces corresponding to the various electronic states involved.²⁸ However, as pointed out by Buchanan and Michl, to generate a full potential energy surface for the six interchromophore degrees of freedom alone would require millions of calculations with only a moderate number of grid points per coordinate.²⁶ Interrogating internal coordinates only exacerbates this problem. Nevertheless, insight can be gained by inspecting a reduced set of coordinates for small model chromophores, as in the theory of (non)adiabatic electron transfer, which is usually illustrated using a single coordinate that absorbs geometry changes in the donor, acceptor, and solvent.

The quantum theory of connected potential wells is attributed to Hund,²⁹ and the concept was revived by Hush to describe chemical reactions.^30,31 In this work, we show that the SF and TTA processes can be conceptually condensed into one intramolecular coordinate, which interconverts between the equilibrium geometries of the singlet and triplet pair states, and representative intermolecular coordinates which modulate the coupling between these states. We make the link between the Marcus-Hush theory of electron transfer^30–40 and singlet fission.⁴¹ Indeed, singlet fission can be formally viewed as an exchange of two electrons, with the charge transfer state behaving as a virtual or real intermediate state.^26,42,43 By plotting the adiabatic electronic energy as a function of the intra- and intermolecular coordinates, one can view SF or TTA as a set of trajectories which sample the regions of the surface variously described by the singlet, triplet pair, or excimer states.

To place the approach on a firm theoretical footing, we begin by calculating adiabatic SF/TTA surfaces for ethylene at a correlated level of electronic structure theory. Ethylene is the simplest π-chromophore and has been invoked as a model (or “toy”) singlet fission system.⁴⁴ We show that the symmetric stretching of the two C=C bonds forms a natural intramolecular coordinate joining minima on the S₁ and ¹(TT) surfaces. This coordinate reveals the connection between singlet fission and Marcus-Hush theory, with the coupling between the diabats modulated by intermolecular coordinates. Aided by density functional theory, the Marcus-Hush treatment of singlet fission is then extended to tetracene and pentacene as prototypical endothermic and exothermic SF systems.

Ethylene calculations were performed using the Firefly program,⁴⁵ which is largely based on the GAMESS program.⁴⁶ Ground and triplet state geometries were obtained at the B3LYP/6-311G(d,p) level. The triplet geometry was then used as a reference, there being little difference in bond angles (see Table I).

The C=C bond length (R in Fig. 1) was scanned from 1.25 to 1.69 Å, and electronic states were calculated using the two highest-energy occupied triplet state ROHF/6-31G(d) orbitals as an active space. Energies were refined using the XMCQDPT routine, which is a second-order perturbation theory.⁴⁷ Three states were considered: The singlet ¹A_g (ground) and ¹B_1u states, and the ³B_1u state, the latter two being π–π^* excited states.

The dimer was first considered at a cofacial distance of 10 Å, in D_2h symmetry (z = 0, D = 10 Å in Fig. 1). The electronic states were calculated as for the monomer but using the four highest-energy orbitals of the quintet state as the active space. The quintet state of the dimer is the lowest-energy state calculable by single-reference self-consistent field methods in which all four frontier orbitals are occupied.

At this geometry, there are two ¹A_g states, one that corresponds closely to the energy of two monomer ground states and one that corresponds to the energy of two monomer ³B_1u triplet states. There are two excited singlets derived from the ¹B_1u monomer excitation. These are the excitonically coupled ¹B_1u and ¹B_3g states, the lower of which is the optically dark ¹B_3g state. The quintet state is near degenerate with the singlet-coupled triplet pair (2¹A_g).

The coupling and emerging energy landscape of these electronic states was then explored as a function of various geometric parameters.

The optimum geometries of the S₀, T₁, and S₁ states of tetracene and pentacene were calculated at the (TD)-B3LYP level of theory using the 6-311G(d) basis set in Gaussian 09.⁴⁸ The reorganization energies for these states were calculated at the optimized geometries of the other states.

The [2,2] XMCQDPT/6-31G(d) potential energy curves of the monomer as a function of C=C bond length (with triplet equilibrium bond angles) is shown in Fig. 2. As expected for π–π^* excitations, the equilibrium bond lengths in the triplet and excited singlet states are longer than in the ground state. The energies are not intended to be accurate: This is not our purpose here. That the triplet is about half the energy of the singlet at this level of theory allows us to investigate the adiabatic singlet fission (and triplet-triplet annihilation) surface of the dimer. For reference, the vertical excitation of ethylene is estimated to be 7.8 eV,⁴⁹ and the present wavefunction is likely to be more compact than calculated using more advanced methods.⁵⁰ 

At an interchromophore distance, D, of 10 Å, the electronic states of the D_2h ethylene dimer are shown as a function of R, the C=C bond lengths, in Fig. 3. The excitonic splitting can hardly be resolved at this distance, with the bright 1¹B_1u and dark 1¹B_3g states being superimposed. The equilibrium bond length of these states is in between that of the monomer 1¹B_1u state and the ground state.

Both excitonically coupled states cross the 2¹A_g state, which corresponds to a triplet state on each monomer. As such, the surface crossing between the 1¹B_1u/1¹B_3g states and the 2¹A_g state represents the first step in singlet fission.²⁸ The highest states shown are those arising from charge transfer.

On the right of Fig. 3, the monomers are brought into van der Waals contact, with an interchromophore distance of 3.4 Å. Here, there is a clear excitonic splitting between the states derived from the 1¹B_1u monomer state, and these have been relabeled S₁ and S₂ (the ground state is relabeled S₀). The 2¹A_g state is relatively unperturbed by the increased contact and has been relabeled ¹(TT). As is well known, there is no coupling between the ¹(TT) and the S₁ states where the chromophores are perfectly overlapped, as in the D_2h geometry.^26,44 Indeed, they are of differing symmetry representations. To couple these states, we must slip the monomers with respect to one another, reducing the symmetry from D_2h to C_2h. In many electron and charge-transfer problems, where the energy or charge is localized to one moiety, it is the asymmetric stretch which couples the two diabatic states. Here, the dimer as a whole changes electronic state, and despite the symmetric nature of the moieties involved, the asymmetric stretch does not couple the relevant surfaces. Indeed, displacement along the asymmetric stretch reduces the symmetry to $C2v$⁠, and the two excited singlet states correlate with B₂ symmetry, while the ¹(TT) state remains as the totally symmetric A₁ representation. As such, this coordinate mixes the B_1u and B_3g singlet states but does not promote singlet fission or triplet-triplet annihilation.

On reducing the symmetry of the dimer, the S₁ and ¹(TT) states couple, both now being of the totally symmetric representation, A_g, in the C_2h point group. A surface plot of the electronic energies as a function of the C=C stretching coordinate R and the slipping coordinate z is given in Fig. 4. The crossing which is permitted at z = 0 becomes an avoided crossing where z ≠ 0, resulting in a conical intersection, analogous to the treatment of the Jahn-Teller problem by Öpik and Pryce.⁵¹ 

While the coupling between the S₁ and ¹(TT) states vanishes at z = 0, the excitonic coupling is maximized, leading to a deep minimum on the S₁ surface. This is the excimer state. Moving along z from the conical intersection, the coupling between the two states results in an avoided crossing with respect to R. As shown by Michl, the S₁–¹(TT) coupling at D = 3.0 Å is maximized in ethylene for z displacements of about 1 Å.^26,44 At large displacements along z, the excitonic coupling reverses the sign as the dimer transitions from H-coupling to J-coupling. This pushes the S₁ state up (it is now above S₂).

At a slip displacement of z = 1 Å, the adiabatic potential energy surfaces are shown as a function of the intramolecular coordinate R and the intermolecular coordinate D in Fig. 5. At large intermolecular distances D, the curves with respect to R resemble the intersecting parabolas of a Marcus diagram. As the S₁–¹(TT) coupling is increased with decreasing D, the surfaces exhibit a pronounced avoided crossing. At very short intermolecular distances, the lower surface exhibits a deep minimum connected to the excimer state.

The potential energy curves plotted above show that singlet fission and triplet-triplet annihilation can be viewed, in the adiabatic picture, as trajectories which circumvent the conical intersection in Fig. 4 and surmount any barrier presented in Fig. 5. The excimer state presents a perilous trap.

Without delving into the geometry-dependence of the various integrals which give rise to the coupling between the S₁ and ¹(TT) states,^26,44 we can explore the overall distance-dependence of the coupling by modeling the calculated Marcus parabolas. At long intermolecular distances, where the coupling is negligible, the ab initio potential energy curves $Vαd$(R; D > 5 Å) [α = S₁,¹(TT)] were taken as diabatic curves and fit as cubic polynomials in R. The ab initio adiabatic curves at shorter distances were then modeled as the interaction between these diabats, offset by $ΔS1(D,z)$ and $Δ(TT)1(D,z)$ from their uncoupled energies: The calculated adiabatic curves were fit to the eigenvalues of the matrix

by adjusting V_SF, $ΔS1$ and $Δ(TT)1$⁠. The resulting geometry-dependence of V_SF, the coupling between the S₁ and ¹(TT) diabats, is illustrated in Fig. 6. The interchromophore distance D-dependence is plotted on a logarithmic scale for z = 1.0 Å. Its linear appearance demonstrates that the coupling is exponentially dependent on distance, as expected for a process mediated by Dexter-like energy transfer mediated by orbital overlap,⁵² but the range is somewhat surprising given the small Gaussian basis employed. The slope of the plot is −0.96 Å⁻¹ on a decadic scale which gives a characteristic decay distance of 0.42 Å. The coupling falls below the thermal energy k_BT at about D > 4.25 Å, implying that beyond this distance, nonadiabatic dynamics needs to be considered. The slip coordinate z-dependence was calculated at D = 3.4 Å, the canonical van der Waals distance, and is plotted in Fig. 6. There is no coupling at z = 0.0 Å for symmetry reasons,^26,44 and the peak is at z = 1.0 Å, as shown by Michl and co-workers.^26,44

From the detailed calculations on ethylene, one obtains a picture of singlet fission which can be applied more broadly. Tetracene and pentacene are widely studied singlet fission materials and are, respectively, endothermic and exothermic prototypes. While ethylene was found to be in the “Marcus normal” regime at the level of theory applied above, it is of interest to see if tetracene and pentacene are in the normal or inverted regime.³⁸ 

We performed (TD)-B3LYP calculations using the 6-311G(d) basis to estimate the energies of the S₀, T₁, and S₁ states at geometries, respectively, optimized for these three electronic states. The resulting energies are reported in Table II. While the B3LYP functional may not be the best choice for absolute excitation energies,^53,54 of importance here are the shapes of the potential energy surfaces of the individual states and not the absolute energies. TD-B3LYP has been shown to accurately reproduce excited state frequencies and therefore force constants.⁵⁵ In ethylene, we considered a single coordinate that connects the minima on the S₀, T₁, and S₁ surfaces—the C=C stretch. For the acenes, we calculated the vectors connecting the minima on the S₁ and T₁ surfaces to the minimum on the S₀ surface, $RS0−T1$ and $RS0−S1$⁠. The scalar product of the normalized vectors, $RS0−T1⋅RS0−S1/|RS0−T1‖RS0−S1|$⁠, was found to be 0.985 for both molecules, corresponding to an angle of just 9°. The ratio of the magnitudes of these vectors, $|RS0−T1|/|RS0−S1|$⁠, was found to be 1.075 and 1.145 for tetracene and pentacene, respectively. As such, the minima of the T₁ and S₁ are similar in magnitude and direction from the S₀ state, which is justified on the basis of orbital occupancy leading to similar differences in bonding character in both excited states. The energy of the S₀ state at the S₁ minimum geometry was found to be about 0.67 of that at the T₁ geometry for both molecules, with pentacene being a smaller energy difference on account of the greater spatial distribution of the excitation. Indeed, the ratio of the reorganization energies in tetracene and pentacene is approximately 4:5 (0.83).

In the interest of simplicity, we investigated the energies of acene dimers along the $RS0−T1$ coordinates of both molecules. The potential curves are approximated as parabolas, and the force constants are calculated from Table II. The offsets of the minima of the electronic states are taken from experiment,^10,56 which assumes non-negligible excitonic coupling between the moieties when one is excited to S₁ so that the global minimum on S₁ lies near the $RS0−T1$ coordinate. The slices along $RS0−T1$ for tetracene and pentacene are shown in Fig. 7. Tetracene is an endothermic fission material, but from Fig. 7, it appears that there is little in the way of an extra barrier to singlet fission due to reorganization. The curve for the singlet state cuts through the minimum of the triplet pair state. As such, one could expect recombination of triplets to be very fast. Pentacene, by contrast, appears poised for singlet fission. The curves for pentacene show that there is barely a barrier to fission, if at all. The present estimate of 3 meV is below the expected accuracy of the model and could easily be diminished by a coupling between the two surfaces. Pentacene is known to undergo fission in just 80 fs.⁵⁷ 

The calculated optimum geometries of the acenes demonstrated that the $RS0−T1$ coordinate lies near parallel to the $RS0−S1$ coordinate and is of a similar length. In the following, we reduce these coordinates to one, which we denote x. The energy of the S₀ state is then approximated by $ES0=λx2$ and the excited singlet by $ES1=λ(1−x)2+ES10$⁠. The energy for a dimer, in which one moiety is in S₁ and the other in S₀, in the absence of excitonic coupling is

The minimum of the S₁ + S₀ curve is thus at x = 1/2. For the curves plotted in Fig. 7, the minimum of the S₁ + S₀ surface is found to be close to 0.5 but differs due to differing reorganization energies in the S₀ and T₁ states. Excitonic coupling will lower the minimum energy, as experimentally observed. The energy of the triplet pair state is

Plotting these curves, one obtains the Marcus diagram shown in Fig. 8.

In the crystalline state, the relative disposition of the monomers is relatively fixed. In this case, one could estimate the coupling between the electronic states by application of the Levich-Dogonadze formula for the transfer rate,^59–63 which we modify for the present formulation, where the reorganization energy in the dimer is λ/2 (Fig. 8),

From the parameters in Table II, one obtains, for tetracene, λ = 0.27 eV. With ΔE_SF = 0.17 eV and V_SF = 0.02 eV, Eq. (4) predicts a prepared-state lifetime of 180 ps, which corresponds well with the single crystal rates determined by Piland and Bardeen.¹⁴ Application to pentacene with λ = 0.23 eV, ΔE_SF = −0.11 eV, and V_SF = 0.02 eV, one predicts a singlet lifetime of 51 fs, which is close to that experimentally observed. Given the approximate nature of the present formulation, one can be satisfied that the Marcus approach is at least semiquantitative and that the coupling V_SF is of the order 0.02 eV in acene crystals. However, caution is warranted: There is an upper limit for the rate as the system transitions from a nonadiabatic to an adiabatic process. Yost et al. found that for pentacene derivatives, deviation from Marcus behavior was observed above V_SF = 0.02 eV.⁴¹ 

Hush predicted the appearance of intervalence charge-transfer bands based on a vertical transition from the initial state.^36,37 Figure 8 suggests that an analogous spectroscopic transition may be observable in endothermic singlet fission systems. This would have a peak at ΔE_SF + λ/2, where λ is the monomer S₀ − T₁ reorganization energy. In tetracene, ΔE_SF = 0.17 eV. From the data in Table II, we find that λ = 0.27 eV and thus the vertical singlet fission transition is expected at hν = 0.31 eV, or 2460 cm⁻¹. This transition may be observable in transient absorption experiments, or activated in a femtosecond pump-push-probe experiment. From the ¹(TT) state, it may be possible to observe the transition back to the S₁ surface, which has an energy of about λ/2 −ΔE_SF. In pentacene, this would be at about 2600 cm⁻¹.

Invoking ethylene as a model system, a curve crossing between the S₁ and ¹(TT) potential energy surfaces is evident as the C=C bonds are symmetrically stretched from the ground state equilibrium geometry. The potential energy curves along this coordinate resemble Marcus parabolas which exhibit an avoided crossing when the monomers are slipped relative to each other. The matrix element which couples the two curves is exponentially dependent on the interchromophore distance and exceeds k_BT for D ≲ 4.25 Å at the optimum slip displacement. The coupling results in a deep excimer well at small interchromophore distances.

DFT calculations support the concept of an intramolecular singlet fission coordinate which links the minima on the dimer S₀ (x = 0) and ¹(TT) surfaces (x = 1) in acenes. The minimum on the dimer S₁ surface is at approximately x = 0.5. It was found that pentacene is near optimally poised for singlet fission from a Marcus perspective, with tetracene disposed to triplet-triplet annihilation. Inspection of the generalized Marcus diagram for endothermic fission reveals a possible vertical transition from the S₁ state which would result in vertical excitation to the ¹(TT) surface.

This work was supported by the Australian Research Council (Centre of Excellence in Exciton Science Grant No. CE170100026).

This paper is dedicated to the memory of Professor Noel Sydney Hush, who was an inspiration to the author.