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STUART A. RICE, PhD, received his bachelor's degree from Brooklyn college and his master's degree and doctorate from Harvard University. He was a junior fellow at Harvard for two years before joining the faculty of The University of Chicago in 1957, where he is currently the Frank P. Hixon Distinguished Service Professor Emeritus.
AARON R. DINNER, PhD, received his bachelor's degree and doctorate from Harvard University, after which he conducted postdoctoral research at the University of Oxford and the University of California, Berkeley. He joined the faculty at The University of Chicago in 2003.
Electronic Structure and Dynamics of Singlet Fission in Organic Molecules and Crystals
Timothy C. Berkelbach
Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, IL, 60637, USA
- I. Introduction
- II. Electronic Structure of Low-Lying Excited States
- A. Weak-Coupling Configuration Interaction Theory
- B. More Accurate Wavefunction-Based Methods
- C. Mechanisms for Singlet Fission
- III. Measuring Charge-Transfer Character
- IV. Charge-Transfer Implications for Singlet Fission
- A. High-Energy CT Configurations and the Superexchange Picture
- B. Low-Energy CT Configurations and Physical Mixing
- V. Theory of Spectroscopy, Reaction Rates, and Singlet Fission Dynamics
- A. The Electronic-Vibrational Hamiltonian and Reduced Dynamics
- B. Validating the Hamiltonian Through Spectroscopy
- C. Rate Theories
- D. Full Quantum Dynamics
- VI. Conclusions and Outlook
Classic work has laid the foundation for our modern understanding of molecular excitons [1-6]. In this sense, much of the phenomenological theory is quite mature and leads to a satisfactory understanding of electronic interactions, as well as the important role played by molecular vibrations and crystalline phonons. And yet, these materials continue to provide fertile ground for new research, which is perhaps a testament to their genuinely complex optoelectronic properties. In general, this chapter is concerned with the renewed interest in a photophysical phenomenon known as singlet exciton fission (defined in the following). The recent intense study of this specific problem has prompted the field to revisit classic topics with modern tools and motivations.
On the experimental side, ultrafast time-resolved and nonlinear spectroscopies in particular have allowed for a richer and more detailed understanding of excited-state dynamics in a host of material systems, including not only organic molecules and crystals but of course also gas-phase molecules, liquids, nanocrystals, and light-harvesting complexes. On the theoretical side, modern computational tools are enabling predictive calculations that can in some cases supersede the semiempirical and phenomenological calculations that were necessarily employed to establish the field. Time-dependent density functional theory (TD-DFT) [7, 8], Green's function-based many-body perturbation theory [9, 10], and the density matrix renormalization group [11, 12] are just three examples of relatively new and powerful tools that are being brought to bear on the electronic structure of organic molecules and crystals. Techniques and capabilities of quantum dynamics, in particular related to reduced density matrix techniques, have also only more recently evolved to produce nonperturbative results for large, multichromophore systems.
The recent interest in organic materials in particular has been driven by a number of potential applications including organic solar cells, light-emitting diodes, and field-effect transistors. From a practical point of view, the advantages of organic materials are twofold. First, the raw materials are cheap and robust, ideally requiring no heavy atoms or special handling. Second, chemical functionalization is mature and should enable for precise control of structural, electronic, and optical properties. Although these advantages have always been recognized, it is only in recent years that such materials have really been employed in consumer technologies. Most relevant, the pressing need for clean energy has encouraged new efforts toward cheap and efficient solar cells. The organic-based solar cells have always trailed their inorganic counterparts in efficiency (admittedly, at lower cost), but "unconventional" light-harvesting technologies might help close that gap. In this vein, the phenomenon of singlet exciton fission has captured the attention of many scientists.
Singlet exciton fission (henceforth, "singlet fission") is a version of carrier multiplication or multiple exciton generation but is unique to the organic semiconductors. Unlike the inorganic semiconductors, organics exhibit a large electron-hole exchange interaction, which is responsible for low-energy triplet states. In a single molecule, the transition from an excited singlet state to a triplet state is spin-forbidden (intersystem crossing) and, therefore, slow unless mediated by strong spin-orbit interactions. However, when two molecules are brought together, a new spin-singlet state is born, which has the character of a triplet excited state on each molecule - that is, it is a multiexciton state. This multiple-excitation character leads to a small oscillator strength and so the state is spectroscopically dark (in linear order). But for sufficiently low-energy triplets, the multiexciton energy may fall within the manifold of low-lying bright singlets and configuration interaction (CI) coupled with nuclear rearrangements could act to populate the multiexciton state following photoexcitation.
Because all involved states are of spin-singlet character, there is reason to believe that the singlet fission process could be fast (compared with fluorescence, intersystem crossing, and other nonradiative recombination mechanisms). If, on a longer timescale, this multiexciton singlet state evolves into some (non-spin-pure) state representing separated triplets, then multiple exciton generation has been achieved: a single photon has produced two (triplet) excitons. With an appropriate tandem or sensitization strategy, singlet fission can improve solar cell efficiencies and even (in principle) surpass the Shockley-Queisser limit [13-15].
The possibility of singlet fission was first discussed in 1965, by Singh et al., while investigating the delayed fluorescence of anthracene . The suggestion was motivated as the reverse process of triplet-triplet (TT) annihilation to generate emissive singlets, which had been recently observed and investigated [17-19]. A few years later in 1968, Swenberg and Stacy invoked singlet fission to explain the quenched fluorescence yield in tetracene crystals . Even in this very early proposition, the authors recognized the potential importance of the so-called charge-transfer (CT) configurations, which had only recently been highlighted in the context of molecular crystals by Rice et al. [2, 21, 22]. Borrowing their theoretical estimates of the relevant matrix elements, energy differences, and the density of states, Swenberg and Stacy performed a golden rule evaluation of the singlet fission rate and found - s or -25 ps. This timescale is significantly shorter than the fluorescence lifetime of the smaller acenes and thus gave credence to the notion that singlet fission was the dominant relaxation pathway for photoexcited singlet excited states in tetracene (ultrafast time-resolved spectroscopy would later show the singlet fission time constant in tetracene to be on the order of 10-100 ps [23-26]). The singlet fission proposal would quickly be verified via magnetic field effects, which unambiguously implicate intermediate triplet states [27, 28]. Subsequent theoretical work was focused on kinetic models of the process, including the interplay between singlet fission, triplet diffusion, and pairwise annihilation [29, 30].
As discussed earlier, singlet fission was largely forgotten for 35 years until it was revived in the context of solar energy conversion [14, 31]. The subsequent 10 years, and especially the most recent 5 years, have seen a flurry of activity aimed at the investigation and characterization of various singlet fission materials. In general, materials systems of interest can be broken up into covalently bound dimers [32-37], thin films, and single crystals [12, 24, 25, 38-42], and more recently into solution , polymers [44, 45], and nanocrystals .
Although I will occasionally make reference to recent experiments, this chapter is about the theoretical and computational description of excitons and their dynamics, in organic molecules and crystals with a focus on singlet fission in the oligoacenes. More specifically, this work aims to connect theoretical results published over many years and in many different fields. Ultimately, I hope to demonstrate a (perhaps surprising) degree of consistency and harmony, the recognition and understanding of which should help advance the field toward new and challenging problems. A number of other reviews on singlet fission have recently appeared, which are less theoretically oriented than the present one [15, 47-49].
The layout of the chapter is as follows. First in Section II, I introduce the weak-coupling CI theory of low-lying states in organic molecules such as the oligoacenes and make connections to more accurate computational techniques. This overview establishes the electronic structure language relevant for singlet fission and introduces the notion of CT configurations, whose importance was recognized very early on in the field of molecular excitons. In Section III, I discuss the difficulties and techniques associated with the quantification of CT character in low-lying excited states. Having established the generic presence of CT states, I discuss the implications for singlet fission in Section IV. This leads to a discussion of...