A. Weak-Coupling Configuration Interaction Theory

In the modern context of singlet fission, this minimal active space approach was first presented by Smith and Michl for the case of two molecules [15], and by myself, Hybertsen, and Reichman for the case of clusters and crystals [52]; however, the model and its ingredients are of course very common and have been used for a variety of problems related to excitons in organic chromophores [17, 22, 53–56]. The theory is simple and intentionally so. Although quantitatively inaccurate (discussed later), this formalism immediately exposes the purely electronic aspects of singlet fission, including two mechanistic “pathways.” For two interacting molecules, the simple theory identifies five potentially low-lying electronic (spin-adapted) configurations, as shown in Fig. 2. I use the notation , where denotes the electronic character of monomer in the dimer wavefunction. The two intramolecular Frenkel excitations (FEs), and , have one molecule in the first singlet excited state , while the other is in the ground state . The intermolecular CT (or ion-pair) excitations, and have one molecule in a cation state C and the other in an anion state A. The TT double excitation is a correlated triplet-pair or multiexciton state, with both molecules in the lowest triplet state . I discuss the quantum mechanical properties of these states in order in the following.

Intramolecular Frenkel-type excitations: The two intramolecular excitations form the starting point of conventional theories of molecular excitons. Indeed, in many molecular crystals, these states constitute a sufficient basis and comprise Frenkel's theory of excitons [57, 58].

Within the HOMO–LUMO picture, the excitation energy (i.e., the diagonal matrix element) of one such configuration is

1

where and are the orbital energies of the LUMO and HOMO of molecule A and two-electron integrals are given by

2

(for simplicity, I assume real orbitals). Physically, the excitation energy is the “bare” orbital energy difference (i.e., the charge or band gap, which can be related to the IP and EA) lowered by the attractive electron–hole Coulomb interaction and increased by twice the electron–hole exchange interaction . These latter two “corrections” to the orbital energy difference can be thought of as a single-molecule exciton binding energy. For future reference, note that the energy of the single-molecule triplet is identical, but has no exchange repulsion,

3

The interaction between intramolecular excitations is given by

4

The aforementioned matrix element physically represents the Coulomb interaction between each molecule's transition charge density, . For large intermolecular separation, a multipole expansion yields as the leading-order term a transition dipole–dipole interaction of the Förster form,

5

where the transition dipole moment is given by

6

Using only the many-body basis of intramolecular excitons, the eigenstates have a dispersion with bandwidth proportional to (details depend on the lattice). For a periodic system and rigid lattice, these FEs are completely delocalized in space but have a vanishing electron–hole separation. However, at least two important mechanisms can act to localize FEs: energetic disorder (especially in low dimensions) and coupling to molecular vibrations and phonons. With increasing localization, the exciton dynamics will exhibit a crossover between band-like and hopping transport [59, 60].

Intermolecular CT excitations: The two intermolecular excited configurations arise from the excitation of an electron from the HOMO of one molecule to the LUMO of the other. These individual CT configurations have a nonvanishing dipole moment. However, for certain highly symmetric assemblies, the eigenstates (after CI) will exhibit a mixture of equal but opposite CT configurations and thus possess a vanishing dipole moment (see Section III). For this reason, it may be more transparent (but formally identical) to work with the “charge-resonance” basis states [61], .

The excitation energy of a CT configuration is approximately given by

7

where the intermolecular exchange integral has been neglected [21, 52]. The CT exciton binding energy is thus equal to the Coulomb integral . Because the CT electron–hole pair separation is larger than that of the intramolecular excitation, the former has a smaller Coulomb attraction, and thus . However, especially in larger acenes and crystalline environments, it can happen that , that is, the exciton binding energies (and thus the total excitation energies) of the intramolecular and CT excitons are approximately the same. In this limit, the two classes of states will mix, as was first predicted by Rice et al. for triplet [21] and singlet [2] excitons in aromatic molecular crystals. Equation (7) can clearly be compared with the classical CT energy expression

8

The CT configurations considered here are only for nearest-neighbor pairs; in principle, non-nearest-neighbor pairs could also be included, which would allow for larger exciton sizes and coupling into the manifold of dissociated exciton states (i.e., free electron–hole pairs). In principle, singlet fission to create triplets must out-compete exciton dissociation (among other radiative and nonradiative processes); however, the exciton binding energy in organic molecular crystals is typically much larger than thermal energy and so exciton dissociation (in the bulk) is quite rare. Nonetheless, within this generalization, there is a clear similarity to the Wannier theory of excitons [58, 62, 63]. Although the latter is conventionally applied to inorganic semiconductors with high dielectric constants, the increasing dispersion and polarizability of larger acenes necessitates this first-order Wannier-like addition to the localized basis of FE states.

Triplet-pair excitations: Finally, the target configuration of singlet fission (on the ultrafast timescale) is the multiexciton triplet-pair state . When two triplet () excitations interact, they produce nine spin states: five quintets, three triplets, and one singlet. Only the latter couples significantly with other singlets, because the electronic nonrelativistic Hamiltonian is spin-preserving. Only the weaker spin–orbit terms will mix states of different spin multiplicity. Again, while this is not important at short times, it is crucial for the proper theoretical description of separation into individual triplets [15, 64, 65]. When properly spin-adapted, the spin-singlet triplet-pair state is given by

9

In principle, nothing prohibits the triplets in this state from occupying non-nearest neighbor molecules. Indeed, triplet diffusion from the nearest-neighbor pair to larger separations is surely required to realize the separation into independent triplets, although biexciton binding and entanglement effects are important topics that have only received minimal attention [66]. However, the reverse process of singlet fission, TT annihilation, has a longer history in the context of light-emitting diodes and (more recently) photon upconversion in solar cells. A theoretical picture of TT annihilation clearly requires a unified description of non-nearest neighbor triplet pairs, triplet exciton diffusion, and the full spin manifold.

To lowest order, the energy of the singlet multiexciton triplet-pair state is the sum of the single-molecule triplet energies, that is, for a homodimer. These simple considerations have led to one of the earliest design rules for singlet fission chromophores, [15], which is a powerfully simple notion. Nonetheless, such a picture neglects the importance of the electronic coupling between bright singlet and dark triplet-pair states, as well as the important role played by vibrations and phonons. These latter finite-temperature effects can ultimately facilitate endoergic (uphill) singlet fission (as in tetracene) or even inhibit strongly exoergic (downhill) singlet fission (as in hexacene) – see Section V.

On the use of diabatic states: In the simplest sense, the relative energies and interactions between configurations dictate the relevant character of the low-lying electronic eigenstates. For fixed nuclei, the configuration basis and the eigenstate basis are trivially related; as the nuclei move, the matrix elements in the configuration basis will change, causing a concomitant change in the electronic character of the eigenstates. The configuration basis is therefore an approximate diabatic basis, that is, a basis whose members have a fixed electronic character (in this case, essentially by construction); the eigenstate basis at each nuclear configuration is precisely the adiabatic basis. For technical reasons [67], only the latter can be rigorously defined; however, the diabatic basis constitutes a conceptual cornerstone in modern chemical and condensed matter physics and a variety of approximate diabatization techniques are well-known in the literature [67, 68]. In principle, the adiabatic basis carries the same information (in the context of singlet fission, see the appendix of Ref. [69]), and the quantum dynamics mediated by the nonadiabatic coupling matrix elements could be simulated by surface hopping [70, 71], quantum-classical [72], or wavepacket-based methods [73]. However, in addition to its clearer interpretation, the diabatic basis is also the simplest one in which to formulate simple rate theories and more complex quantum dynamics, as discussed in Section V. For these reasons my coauthors and I advocated for the development of diabatic approaches to singlet fission electronic structure and dynamics in our first report, Ref. [52]. Later works demonstrated the ability of constructing more accurate dimer Hamiltonians in the diabatic basis, as discussed in the next section.

Another important aspect of the localized diabatic basis concerns the approximate construction of Hamiltonians for larger aggregates. Somewhat analogous to fragment-based approaches in quantum chemistry, an aggregate Hamiltonian can be straightforwardly constructed from potentially high-accuracy dimer Hamiltonians, as discussed in the next section. Of course, the subsequent electronic structure will neglect important environmental effects due to delocalization and screening. As such, this approach requires some adjustment of the energies and couplings that enter the final aggregate Hamiltonian [69]. Comparison with crystalline band structure and linear spectroscopy (see Section V) provides an important test of this procedure.

B. More Accurate Wavefunction-Based Methods

As warned earlier, the direct implementation of the minimal active space CI method is expected to yield poor excitation energies. Therefore, higher accuracy wavefunction-based methods have been more recently employed for the study of singlet fission. The excited-state electronic structure of the acenes is especially challenging because it (ideally) requires a combined treatment of static correlation (i.e., multireference character) in the valence orbitals as well as dynamical correlation (i.e., orbital relaxation and beyond) in the remaining orbitals. For molecular dimers, the restricted active space two spin-flip method [74, 75], which has been applied to singlet fission [76, 77], targets the same HOMO–LUMO active space with single excitations into and out of the active space to allow for orbital relaxation and improves upon the energies of the simple CI treatment described earlier. Higher accuracy requires larger active spaces and dynamical correlation. Very early work on singlet fission in pentacene [78] employed such a method (CASSCF, followed by multireference perturbation theory), which was also used and refined by Ananth et al. in their study of the low-lying states of pentacene dimers with crystal-like geometry [79].

The philosophy of few-state, low-energy effective Hamiltonians has a long history in quantum chemistry [80, 81] and is also embodied in semiempirical Hamiltonians, such as the Pariser–Parr–Pople model of conjugated molecules [55]. In such procedures, the one- and two-electron matrix elements in a small model subspace should be understood as ones that incorporate screening effects, which have been “folded in” from excitations to and among the neglected orbitals. More modern, first-principles approaches to construct effective Hamiltonians include canonical transformation approaches [82, 83] and the active space decomposition of Shiozaki et al. [84, 85]. The latter approach in particular aims to directly construct a many-body diabatic basis, without diabatization from an adiabatic calculation, and represents a powerful ab initio approach to generating chemically intuitive Hamiltonians. A similar approach to diabatic states via configuration restrictions has recently been given by Casanova and Krylov [86] and one based on spin and charge constraints to density functional theory was used by van Voorhis et al. [41].

Two important conclusions (regarding the electronic structure) emerge from the work of Ananth et al. [79] and Shiozaki et al. [85]: (i) the HOMO–LUMO active space provides an excellent qualitative and in some cases quantitative description of the excited-state structure of the tetracene and pentacene dimers; (ii) correlated diabatic wavefunctions can be directly constructed or obtained via diabatization of high-quality adiabatic wavefunctions to yield a highly accurate five-state basis, which is physically like the one described earlier. These two observations justify the use and explain the success of the minimal CI picture [15, 47, 52, 87]. Remarkably, Shiozaki et al. report that the absolute minimal two-orbital-per-molecule active space with no dynamical correlation yields off-diagonal couplings that are only different by 20–25%. Unsurprisingly, the diagonal energies are much more sensitive and are shifted by as much as 0.8 eV. Anticipating this result, my coauthors and I originally calculated the couplings in the simple CI theory (i.e., as one-electron matrix elements) and investigated the dynamical consequences of shifting the diagonal energies [87].

C. Mechanisms for Singlet Fission

At the dimer level, the simple theory espoused earlier suggests two mechanisms for singlet fission under the assumption that the relevant configuration following photoexcitation is an intramolecular Frenkel excitation (or some linear combination of such states). The first mechanism has been called the “direct” one and is enabled by the electronic coupling (CI) between and . While the magnitude of this interaction is highly system dependent, essentially all calculations for unsubstituted acenes with crystal-like intermolecular separations find that this interaction is very small (roughly 1–5 meV) [76, 79, 85, 87, 88].

The second mechanism has been called the “mediated” one, where mediation takes place via CT states. The interest in CT states originates from the observation that their interactions with the other states, that is, and , are large (on the order of 100 meV); however, their energy can be quite high. Therefore, depending on the relative energies of the diabatic states, the participation of CT states can take many different forms, depending in part on the finite-temperature dynamics of the nuclei. In the remainder of this chapter, I focus on the CT component of this story, including how the mediated mechanism has been studied and verified. There will no doubt be singlet fission systems for which the “direct” mechanism is dominant; two anticipated examples are when the mediated mechanism is symmetry-forbidden and when the chromophores are very far apart. However, the crystalline acenes are classical singlet fission materials, and they exhibit some of the largest fission rates; the evidence is now quite strong that the diabatic CT states are controlling singlet fission in these important prototypical materials [49, 69, 87–89].

A. High-Energy CT Configurations and the Superexchange Picture

For CT states with very high energies, the CT component of the low-lying eigenstates is very small; this limit can lead to a second-order superexchange-like mechanism [52, 87]. In the superexchange picture, one considers the first-order perturbed diabatic wavefunctions, due to the coupling with CT configurations:

12

13

The total interaction between these states then has two parts: a first-order direct coupling and a second-order superexchange coupling [20, 87],

14

This effective interaction is the result of integrating-out the high-energy CT states, to yield a simpler few-state picture of the singlet fission process. It is perhaps worth emphasizing that superexchange contains no physics that cannot be obtained from a full diagonalization of the Hamiltonian into its exact eigenstates. Rather, the superexchange concept is a physical one that qualitatively distinguishes between a small (“virtual”) mixing of CT character and a large (“physical”) mixing of CT character into the low-lying eigenstates. Superexchange describes the former case: to a high accuracy, one can define a new set of diabatic states that excludes the CT configurations while retaining their electronic character; the small CT mixing produces a second-order coupling between the (otherwise weakly interacting) intramolecular excitation and the TT state.

The superexchange picture in singlet fission is almost completely analogous to the case of magnetic interactions in transition metal oxides, where the superexchange idea was first proposed by Kramers [112]. In that case, the same second-order coupling leads to a magnetic interaction between next-nearest-neighbor transition metal atoms. The high-energy CT state is a double-occupancy configuration arising due to CT from the transition metal onto a bridging oxygen atom; the double occupancy is expensive, and so the configuration is high in energy. This picture was further investigated by Anderson [113]. The same exact insight leads to (in a very formal sense) the so-called – model after projecting out high-energy double-occupancy (CT) configurations in the Hubbard model [114]. Since these early works in condensed matter physics, superexchange has been found to be a useful and ubiquitous picture of mediated electronic interactions. A few important examples of superexchange in chemical physics include the electron transfer mechanism of biological donor–bridge–acceptor complexes [115, 116], charge transport through single-molecule junctions [117], and non-Förster excitation energy transfer [54, 56].

Whether a superexchange picture is meaningful obviously depends on the energetics of a given material system. On the basis of a simple CI calculation, my coauthors and I previously advocated that isolated molecular acene dimers and small oligomers would have an energetic ordering that supports superexchange [87]. This finding was strongly corroborated by Shiozaki and coworkers [85], whose active space decomposition yields diabatic CT states, which are approximately 1 eV higher in energy than the FE or TT states. The smaller oligoacenes can also be thought of as exhibiting singlet fission through a superexchange mechanism, because the CT mixing is relatively minor and the excitons are predominantly of Frenkel character. However, the triplet-pair state in crystals of smaller acenes is usually much higher in energy than the low-lying singlets, which precludes efficient singlet fission.

At this stage, note that the superexchange concept only provides an interaction between the purported initial and final states of the (early-time) singlet fission process; it says nothing yet about the rate. A microscopic rate expression can only be derived when considering the finite-temperature dynamics of the nuclei and their coupling to the excited-state electronic structure. This aspect of the problem is discussed in Section V, after a detailed discussion of the electronic aspects of singlet fission in crystals.

B. Low-Energy CT Configurations and Physical Mixing

As discussed earlier, the handful of theoretical treatments for excited states in true crystalline environments indicate an increasingly non-negligible CT character in the low-lying eigenstates. In this regime, superexchange is no longer relevant: the interaction with diabatic CT configurations is nonperturbative. Instead, the intramolecular FE and intermolecular CT configurations are all nearly degenerate and thus strongly interacting, leading to eigenstates with truly mixed character. If one first diagonalizes the “singles” excited-state manifold, that is, the part of the crystal Hamiltonian containing FE and CT configurations, this leads to a set of singlet eigenstates with high oscillator strength, which I simply call , and so on.

Upon photoexcitation near the absorption edge, the system is prepared in some combination of the aforementioned mixed eigenstates, . The fraction of CT character determines the coupling strength with the triplet-pair TT state. Schematically,

15

16

17

In reality, the overall effect is subtle due to crystal geometries and interfering pathways, but the result immediately suggests that the rate of singlet fission increases with increasing CT character in the photoexcited states. Again, I emphasize that this interpretation is only valid for those materials with (i) low-energy CT configurations and (ii) interactions with CT states that are stronger than those between FE and TT. These criteria appear to be satisfied in the larger oligoacene crystals such as tetracene, pentacene, and hexacene [42, 69, 94, 118].

Despite the increased singlet fission coupling, an increased CT character has other implications for the low-energy excited states. The first is related to the earlier discussion, concerning the Davydov shift. It appears to be the case that the energy of CT configurations falls faster than that of intramolecular excitations with increasing acene length. This gives rise to the increasingly larger CT character in the eigenstates, and the concomitant increase in the Davydov shift. The changing eigenstate energy necessarily alters the energy alignment between and TT, which substantially alters the singlet fission rate, as is discussed in Section V.

The second implication of increased CT character is a decreased oscillator strength. The intramolecular oscillator strength is significantly larger than the intermolecular one: the oscillator strength is roughly proportional to the overlap between the HOMO and LUMO involved in the transition. In the highly simplified limit where only intramolecular excitations carry oscillator strength and CT excitations are completely forbidden, the brightness of a given eigenstate depends on the balance of FE and CT character. In other words, the CT component borrows oscillator strength from the bright FE states. With this interpretation, we observe that increasing CT character in low-energy states diminishes the relative oscillator strength. With too much CT character, the states could become essentially dark.

Note that this observation is in fact very analogous to the situation in the Wannier theory of excitons in inorganic semiconductors, mentioned in Section A. In that case, the exciton envelope wavefunction , where is the electron–hole separation, satisfies a hydrogen-like Schrödinger equation, leading to states with angular momentum character s, p, d, and so on. The absorption coefficient can be written as [58]

18

where is the photon polarization, is the single-particle momentum matrix element, and is the exciton energy. The usual approximation is to take to be independent of , that is, completely local in space; this is analogous to assuming that only intramolecular excitations are dipole-allowed. After this approximation, the absorption coefficient becomes

19

leading to the classic result that only s-type excitons are absorbing in linear spectroscopy. All states with , that is, all purely CT excitons, are dark. Even within the s-type manifold, the intensity is directly proportional to the probability of the electron and hole occupying the same site (), which is equivalent to the fraction of on-site (non-CT) excitation character. (A minor caveat: in the Wannier picture, the exciton wavefunction is understood to be coarse–grained over one or more unit cells, so the “local” excitations are not strictly intra-atomic; this is to be expected because the valence electron and hole bandwidth in inorganic semiconductors is roughly an order of magnitude larger than in organics. In any event, the physical picture is the same: local excitations are bright, and nonlocal excitations are dark.)

A. The Electronic–Vibrational Hamiltonian and Reduced Dynamics

To go beyond the static electronic structure picture of excitons requires a time-dependent treatment of the quantum dynamics of singlet fission. For a fully microscopic theory, we employ an explicit consideration of the vibrational degrees of freedom. In particular, we can write the total electronic-plus-vibrational Hamiltonian as

20

Here, is the diabatic electronic Hamiltonian discussed earlier, which can have as few as five states (for singlet fission dimers) or many more states (for clusters and crystals). Quite generally, the vibrational Hamiltonian can be taken as the collective normal modes of molecular vibrations, crystal phonons, and/or solvent polarization fluctuations (all indexed by ),

21

Here, is the vibrational frequency, and are the mass-weighted momentum and coordinate, and and are bosonic creation and annihilation operators. The electronic–vibrational interaction can be written generically in the diabatic basis as

22

The character of excitation energies in the diabatic basis is predominantly “intramolecular,” and so the nuclear dependence of the diagonal terms will be dominated by high-frequency intramolecular vibrations (in the crystal phase). The electronic coupling in the same basis is significantly more “intermolecular,” and so the nuclear dependence of off-diagonal terms will involve low-frequency intermolecular vibrations. In other words, the intramolecular vibrations modulate the “energies,” and the intermolecular vibrations modulate the “couplings”; these are only physically motivated observations, which may be highly system dependent. However, this separation into “Holstein” and “Peierls” couplings seems to hold well for single-particle states and charge transport [119].

At finite-temperature, the statistical ensemble averaging demands use of the density operator, as opposed to the wavefunction. The total density operator , an operator in the electronic-plus-vibrational Hilbert space, obeys the Liouville equation

23

For many (but not all) observables of interest, we are only be concerned with dynamics in the electronic state space, averaged over the nuclear degrees of freedom. This leads to the reduced electronic density operator, obtained by tracing over the vibrational degrees of freedom, . The reduced density operator obeys the equation of motion

24

In the absence of coupling to vibrations, the diagonalization of yields the exact adiabatic eigenstates at the ground-state geometry,

25

This basis is frequently referred to as the “exciton” basis, as opposed to the adiabatic Basis, which diagonalizes the Born–Oppenheimer Hamiltonian at each nuclear configuration. Lowest order perturbation theory in the electron–vibration interaction leads to the well-known (here, time-local) Redfield equations in the exciton basis [56, 120, 121],

26

although other perturbative and numerically exact solutions of Eq. (24) are possible, which is discussed later. Detailed expressions for the Redfield tensor elements can be found in the literature [52, 56], but suffice to say this tensor is responsible for the vibrationally induced population relaxation and decoherence . The density matrix approach provides access to rates through the diagonal population elements, but also provides access to spectroscopies (in the presence of vibrations) through the off-diagonal coherence matrix elements [56].

B. Validating the Hamiltonian Through Spectroscopy

The parameterization or calculation of all vibrational degrees of freedom and their coupling to the electronic excited states is a very demanding program. However, for a given model of the electronic–vibrational interaction, we can compare with experimental linear absorption spectroscopy to validate both the electronic and vibrational treatment in the Hamiltonian.

For single oligoacene molecules, the CC 1400 cm vibration is known to be the most strongly coupled to the electronic transitions. To treat the dynamics in this limit, we only include two electronic states (the ground state and the excited state), coupled via a dipole operator , where denotes the Hermitian conjugate. The vibrational coupling is taken to be diagonal and linear in the normal mode coordinate,

27

where is the exciton–vibration coupling strength and is the dimensionless Huang–Rhys factor. For this description, the time-local Redfield equation, that is, the second-order time-local cumulant, provides the exact solution to this nonperturbative pure-dephasing problem [122]. The so-calculated single-molecule absorption spectrum is shown in Fig. 4(a) and compared with the experimental one taken from Ref. [123]. The classical vibronic progression is clearly seen with a spacing of 1400 cm meV. The vertical excitation energy in this calculation is 2.25 eV and the reorganization energy of 120 meV () is responsible for the lowered absorption onset of the 0–0 transition at around 2.13 eV. It is important to note that this calculation does not require the explicit treatment of the vibrational Hilbert space; the vibrational effects are captured implicitly through the time dependence of the reduced density operator.

When intermolecular couplings are added to the Hamiltonian, the time-local resummation is no longer an exact treatment of linear electronic–vibrational coupling; however, its Markovian limit provides a vibration-induced finite-temperature linewidth. Furthermore, in the crystal phase, phonons, anharmonicities, and other differences in the local environment can also modify the distribution of vibrational frequencies leading to a broadened vibrational density of states (discussed later). By extending the diabatic dimer Hamiltonian to aggregates and crystals, as described at the end of Section IIA, we can perform such quantum dynamics simulations to validate the Hamiltonian via polarized linear absorption [69]. The results are shown in Fig. 4(b) and compared with the polycrystalline experimental absorption from Ref. [40]. Even in the absence of explicit vibronic effects, the results are clearly quite good and capture the main features as well as the sign and magnitude of the Davydov shift, discussed earlier. For pentacene considered here, the Davydov shift is 150 meV, twice as large as in tetracene, indicative of the increased CT character in the low-lying excited states. Recent work from Spano et al. has performed a similar calculation with an enlarged (non-nearest-neighbor) CT exciton basis and explicit vibronic states, which leads to excellent agreement with both components of the polarized absorption [118]. In the absence of vibronic effects, that work finds a spectrum that is very similar to the one shown in Fig. 4(b); the major effect of explicit vibrations is an H-aggregate-like vibronic structure at energies above 2 eV. The combined results shown in Fig. 4 suggest that we have a highly accurate electronic–vibrational Hamiltonian and appropriate treatment of its quantum dynamics.

Simulating the dynamics of singlet fission requires only the addition of the triplet-pair manifold of states. Interestingly, including the coupling to these TT states only slightly changes the absorption because the CT-mediated coupling is not sufficient to overcome the non-negligible energy difference between the bright Frenkel component and the dark TT component.

C. Rate Theories

Before discussing the results of explicitly time-dependent quantum dynamics simulations, we first discuss the applicability of various microscopic rate theories. These rate theories can be derived in perturbative limits of the full quantum master equation, Eq. (24). In any situation where the explicit inclusion of CT states can be neglected – either due to a strong direct coupling, the appropriate use of a superexchange coupling, or when the initial state is understood to be a mixture of Frenkel and CT excitons – we can write the electronic excited-state Hamiltonian in a schematic two-state picture,

28

Again, exact diagonalization will yield the eigenstates; however, we can gain some preliminary insight from a golden rule evaluation of the singlet fission rate constant,

29

The Franck–Condon-weighted density of states depends on the temperature and the detailed form of the coupling to molecular vibrations, phonons, or solvent nuclei. Note that when the physical CT mixing is not large, then the first two contributing terms to the coupling matrix element are precisely those given in Eq. (14). When Eq. (14) is inserted in the rate expression Eq. (29), we find essentially the exact formula given in the early proposal of singlet fission by Swenberg and Stacy [20]. Only more recently have atomistic calculations been performed to more precisely quantify the terms contributing to this rate expression. A particular case of interest concerns the coherent addition of many different terms contributing to the total effective coupling, as in Eq. (14). Each term carries a sign and thus constructive or destructive interference can occur to enhance or suppress the singlet fission process [15, 47, 87].

Historical work has shown that in the limit of large interactions , the rate becomes independent of . This marks a transition from a nonadiabatic rate process to an adiabatic rate process. This limit of large is of course beyond the perturbative limit of the golden rule. Early work in this area was motivated by electron transfer in polar solvents, where the solvent dynamics was assumed to be slow and classical [124–126]. The unification of nonadiabatic and adiabatic dynamics in the presence of a high-frequency intramolecular vibration requiring a quantum mechanical treatment was first achieved by Jortner and Bixon [127]. In the context of singlet fission, this rate equation takes the form

30

The sum is over vibrational energy levels of the high-frequency mode, which occurs at meV (1400 cm) in the oligoacenes. The golden rule (nonadiabatic) rate constant appears in the numerator,

31

and the Franck–Condon-weighted adiabaticity parameter appears in the denominator,

32

Here, is the Huang–Rhys factor of the intramolecular vibration and and are the characteristic frequency (inverse relaxation time) and reorganization energy of the remaining nuclear degrees of freedom. This approach was recently employed by van Voorhis and coworkers [41] to explain the measured singlet fission rate constants in a large class of crystal-phase pentacene derivatives. The derivatives contained side chains of varying size, to ultimately tune the intermolecular separation and thus the singlet fission electronic coupling [41]. The study found that the nonadiabatic to adiabatic transition in pentacene derivatives occurs around meV. The take-home message is crucial: increased coupling is good for fast fission but has diminishing returns beyond a certain threshold.

D. Full Quantum Dynamics

With a large aggregate diabatic Hamiltonian including coupling to the triplet-pair states (totaling up to thousands of quantum states), we have recently performed Redfield dynamics calculations of singlet fission in pentacene [69]. The calculations use the same single-excitation manifold of states that produced the absorption spectrum shown in Fig. 4(b) and similarly parameterized triplet-pair states, with a coupling to vibrations appropriate for the condensed phase. Results of these simulations predict a singlet fission timescale of about 300 fs, which compares quite favorably with the experimental timescale of about 80–120 fs. Minor changes in the Hamiltonian can easily reproduce this timescale, though we choose not to pursue this degree of empiricism. Analysis of the population dynamics shows an overall incoherent kinetics for the total state populations, although individual quantum states localized throughout the crystal show highly oscillatory transient dynamics. For related reasons, different treatments of the exciting laser pulse (i.e., the initial condition) yield interesting and sometimes complex dynamics at very short times in the order of 5–10 fs.

The overall incoherent dynamics can partially be attributed to our treatment of the intramolecular vibrations and crystal phonons. In particular, we employed a coupling to an environmental reaction coordinate, which has a linear response and fluctuation correlations that decay exponentially in time [128, 129]. This is equivalent to linear coupling to a bath of harmonic oscillators characterized by an Ohmic spectral density with Lorentzian cutoff (also known as a Debye spectral density or the overdamped Brownian oscillator model) [130]. Specifically, we employed the diagonal interaction

33

where is the reaction coordinate with fluctuations characterized by the autocorrelation function

34

where and are the reorganization energy and the characteristic vibrational frequency, as mentioned before. For a few-state model of singlet fission, Tao used the same Hamiltonian and performed semiclassical quantum dynamics [131], which in general embody an entirely different set of approximations compared with Redfield theory. However, the results were found to be in excellent agreement, suggesting that both methods are giving an accurate description of the dynamics generated by the model Hamiltonian.

Since our original treatment of the dynamics in models of singlet fission dimers [87], there have been a number of important extensions to other dimer systems, including more accurate treatments of the vibrational coupling. In particular, Beljonne et al. performed a first-principles calculation of the electronic structure and vibrational couplings in pentacene and rubrene dimers and trimers [132]. These authors then used the numerically exact multiconfigurational time-dependent Hartree approach to simulate the singlet fission dynamics. For pentacene, these results corroborated the superexchange-like pathway in small aggregates, and the overall decay dynamics are quite similar to those from Redfield theory but with additional transient vibrational oscillations. In another recent work, Fujihashi and Ishizaki analyze the possibility and implications of correlated vibrational fluctuations within a few-state model of singlet fission [133] and again find that the long-time decay dynamics are essentially unchanged from our Redfield simulations; however, the short-time dynamics can look quite different based on the assumed correlations or anticorrelations in the coupling to the reaction coordinate.

I conclude this section by highlighting two of the nontrivial predictions achieved with the microscopic approach presented here. First, our early work on model singlet fission dimers investigated the validity of the superexchange rate theory embodied in Eqs (14) and (29). Because the Redfield approach is completely nonperturbative in the electronic degrees of freedom (i.e., it works in the electronic eigenstate basis), it is capable of capturing higher order effects in beyond the golden rule. In Ref. [87], we demonstrated that the nonadiabatic rate theory breaks down around meV, and the dynamics simulations show a turnover to reduced singlet fission rates. As discussed earlier, this nonadiabatic to adiabatic transition was later demonstrated with an experimental tour de force covering six pentacene derivatives and two to three orders of magnitude in the couplings and the rates [41]. Those results were interpreted in the framework of the Jortner–Bixon rate theory, Eq. (30), with electronic parameters calculated using constrained density functional theory. The agreement between our simulations and this rate expression is especially encouraging and suggests a consistent physical understanding. Figure 5(a) compares the results of our dynamical theory (using the electronic couplings calculated in Ref. [41]) to the experimental singlet fission rates.

The final example concerns a nonperturbative effect in the vibrational degrees of freedom, rather than in the electronic coupling. In particular, my collaborators and I were motivated to extend the family of oligoacenes exhibiting singlet fission to include hexacene. A synthesis of unsubstituted crystalline hexacene was reported for the first time only recently, in 2012 [134]. The microscopic theory described here predicts strongly exoergic singlet fission like in hexacene to be a multiphonon process. For a model with linear coupling to phonons, the lowest order (Redfield-like) 1-phonon relaxation rate actually predicts a vanishing singlet fission rate. Physically, the large excess energy of singlet fission in hexacene cannot be taken on by a single phonon, because no such high-frequency phonons exist in the material. However, with a lower probability, multiple simultaneous phonon excitations can occur to accommodate this large excess energy. From a theoretical point of view, higher order perturbation theory with linear coupling or lowest order perturbation theory with a nonlinear coupling can both realize this behavior [135]. In Ref. [42], we applied the latter formalism, that is, still using the Redfield framework but with an electronic–vibrational interaction of the form

35

where clearly recovers the conventional linear-coupling expression. With this form, we provided a rate expression for multiphonon singlet fission, and the results are shown in Fig. 5(b) and compared with experimental values for tetracene, pentacene, and hexacene. Via transient absorption spectroscopy, the singlet fission time of hexacene was observed to be 530 fs [42], roughly five times slower than in pentacene, and in good agreement with the prediction of the theory. It is important to recognize that singlet fission in hexacene is not faster than in pentacene simply because it is “more downhill” in energy; this is essentially the physics embodied in exponential energy gap laws [136, 137] and (in a less precise sense) the inverted regime of Marcus theory [138].

The intermolecular interactions and vibrational degrees of freedom mediating singlet fission are relatively similar across the (unsubstituted) acenes, and so the energy difference is (in general) the only remaining material property that can explain the variations in the rate. Indeed, the theory described here ascribes the comparatively slow singlet fission in tetracene to the endoergicity, which also leads to a predicted temperature dependence. Although early studies reported such a temperature dependence [39], later experiments demonstrated temperature-independent singlet fission down to very low temperature [23–25]. However, more recent work from Bardeen et al. has identified a significant difference in the singlet fission behavior of tetracene films and single crystals [48], with the latter showing the expected activated temperature dependence.