Microscopic theory of singlet exciton fission. I. General formulation.

Singlet fission, a spin-allowed energy transfer process generating two triplet excitons from one singlet exciton, has the potential to dramatically increase the efficiency of organic solar cells. However, the dynamical mechanism of this phenomenon is not fully understood and a complete, microscopic theory of singlet fission is lacking. In this work, we assemble the components of a comprehensive microscopic theory of singlet fission that connects excited state quantum chemistry calculations with finite-temperature quantum relaxation theory. We elaborate on the distinction between localized diabatic and delocalized exciton bases for the interpretation of singlet fission experiments in both the time and frequency domains. We discuss various approximations to the exact density matrix dynamics and propose Redfield theory as an ideal compromise between speed and accuracy for the detailed investigation of singlet fission in dimers, clusters, and crystals. Investigations of small model systems based on parameters typical of singlet fission demonstrate the numerical accuracy and practical utility of this approach.

[1]  Alessandro Troisi,et al.  Dynamics of the intermolecular transfer integral in crystalline organic semiconductors. , 2005, The journal of physical chemistry. A.

[2]  U. Manthe,et al.  Wave‐packet dynamics within the multiconfiguration Hartree framework: General aspects and application to NOCl , 1992 .

[3]  U. Manthe,et al.  The multi-configurational time-dependent Hartree approach , 1990 .

[4]  P. Rebentrost,et al.  Atomistic study of the long-lived quantum coherences in the Fenna-Matthews-Olson complex. , 2011, Biophysical journal.

[5]  A. Hese,et al.  Laser spectroscopy of free pentacene molecules (I): The rotational structure of the vibrationless S1←S0 transition , 1998 .

[6]  Klaus Ruedenberg,et al.  Localized Atomic and Molecular Orbitals , 1963 .

[7]  William H. Miller,et al.  Self-consistent hybrid approach for complex systems: Application to the spin-boson model with Debye spectral density , 2001 .

[8]  Hilde D. de Gier,et al.  Explorative computational study of the singlet fission process , 2012 .

[9]  K. Blum.  Density Matrix Theory and Applications , 1981 .

[10]  K. Ghiggino,et al.  Rate expressions for excitation transfer. IV. Energy migration and superexchange phenomena , 1995 .

[11]  A. Leggett,et al.  Dynamics of the dissipative two-state system , 1987 .

[12]  Benjamin T. Miller,et al.  A parallel implementation of the analytic nuclear gradient for time-dependent density functional theory within the Tamm–Dancoff approximation , 1999 .

[13]  J. Luther,et al.  Peak External Photocurrent Quantum Efficiency Exceeding 100% via MEG in a Quantum Dot Solar Cell , 2011, Science.

[14]  Patrick Rebentrost,et al.  Non-Markovian quantum jumps in excitonic energy transfer. , 2009, The Journal of chemical physics.

[15]  Timothy C. Berkelbach,et al.  Microscopic theory of singlet exciton fission. II. Application to pentacene dimers and the role of superexchange. , 2012, The Journal of chemical physics.

[16]  Levi,et al.  Hot-carrier cooling in GaAs: Quantum wells versus bulk. , 1993, Physical review. B, Condensed matter.

[17]  Chandler,et al.  Solving the sign problem in quantum Monte Carlo dynamics. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[18]  D. L. Dexter.  A Theory of Sensitized Luminescence in Solids , 1953 .

[19]  M. Beard,et al.  Highly efficient multiple exciton generation in colloidal PbSe and PbS quantum dots. , 2005, Nano letters.

[20]  M. E. Casida,et al.  Progress in time-dependent density-functional theory. , 2011, Annual review of physical chemistry.

[21]  L. Reining,et al.  Double excitations in finite systems. , 2009, The Journal of chemical physics.

[22]  T. Főrster,et al.  10th Spiers Memorial Lecture. Transfer mechanisms of electronic excitation , 1959 .

[23]  J. Skinner,et al.  Quantum Dynamics and Vibrational Relaxation , 1999 .

[24]  J. Eaves,et al.  A microscopic model of singlet fission. , 2012, The journal of physical chemistry. B.

[25]  C. Bardeen,et al.  The dependence of singlet exciton relaxation on excitation density and temperature in polycrystalline tetracene thin films: kinetic evidence for a dark intermediate state and implications for singlet fission. , 2011, The Journal of chemical physics.

[26]  M. Nakano,et al.  Diradical Character Based Design for Singlet Fission of Condensed-Ring Systems with 4nπ Electrons , 2012 .

[27]  E. Gross,et al.  Time-dependent density functional theory. , 2004, Annual review of physical chemistry.

[28]  S. Louie,et al.  Electron-hole excitations and optical spectra from first principles , 2000 .

[29]  S. Mukamel,et al.  Lindblad equations for strongly coupled populations and coherences in photosynthetic complexes. , 2009, The Journal of chemical physics.

[30]  Paul M Zimmerman,et al.  Mechanism for singlet fission in pentacene and tetracene: from single exciton to two triplets. , 2011, Journal of the American Chemical Society.

[31]  Mark A Ratner,et al.  Singlet fission for dye-sensitized solar cells: can a suitable sensitizer be found? , 2006, Journal of the American Chemical Society.

[32]  Thomas Renger,et al.  On the relation of protein dynamics and exciton relaxation in pigment–protein complexes: An estimation of the spectral density and a theory for the calculation of optical spectra , 2002 .

[33]  N. Makri,et al.  TENSOR PROPAGATOR FOR ITERATIVE QUANTUM TIME EVOLUTION OF REDUCED DENSITY MATRICES. I: THEORY , 1995 .

[34]  J. Ross,et al.  Statistical reduction for strongly driven simple quantum systems , 1978 .

[35]  R. Kubo,et al.  Time Evolution of a Quantum System in Contact with a Nearly Gaussian-Markoffian Noise Bath , 1989 .

[36]  A. Marini,et al.  Double excitations in correlated systems: a many-body approach. , 2011, The Journal of chemical physics.

[37]  Mak,et al.  Rate concept and retarded master equations for dissipative tight-binding models. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[38]  Klaus Schulten,et al.  The effect of correlated bath fluctuations on exciton transfer. , 2011, The Journal of chemical physics.

[39]  Mark A Ratner,et al.  Singlet exciton fission for solar cell applications: energy aspects of interchromophore coupling. , 2010, The journal of physical chemistry. B.

[40]  M. Wasielewski,et al.  Competition between singlet fission and charge separation in solution-processed blend films of 6,13-bis(triisopropylsilylethynyl)pentacene with sterically-encumbered perylene-3,4:9,10-bis(dicarboximide)s. , 2012, Journal of the American Chemical Society.

[41]  D. Reichman,et al.  On the nonperturbative theory of pure dephasing in condensed phases at low temperatures , 1996 .

[42]  Gregory D. Scholes,et al.  Rate expressions for excitation transfer. II. Electronic considerations of direct and through–configuration exciton resonance interactions , 1994 .

[43]  G. Fleming,et al.  Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature , 2009, Proceedings of the National Academy of Sciences.

[44]  A. Nozik.  Spectroscopy and hot electron relaxation dynamics in semiconductor quantum wells and quantum dots. , 2001, Annual review of physical chemistry.

[45]  L. Muñoz,et al.  ”QUANTUM THEORY OF SOLIDS” , 2009 .

[46]  U. Weiss.  Quantum Dissipative Systems , 1993 .

[47]  Francesco Petruccione,et al.  The Theory of Open Quantum Systems , 2002 .

[48]  G. Fleming,et al.  Unified treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: reduced hierarchy equation approach. , 2009, The Journal of chemical physics.

[49]  E. Venuti,et al.  Interaction of charge carriers with lattice and molecular phonons in crystalline pentacene. , 2011, The Journal of chemical physics.

[50]  H. Queisser,et al.  Detailed Balance Limit of Efficiency of p‐n Junction Solar Cells , 1961 .

[51]  N. Makri.  Improved Feynman propagators on a grid and non-adiabatic corrections within the path integral framework , 1992 .

[52]  V. May,et al.  Charge and Energy Transfer Dynamics in Molecular Systems: MAY:CHARGE TRANSFER 3ED O-BK , 2011 .

[53]  Irwin Oppenheim,et al.  Memory effects in the relaxation of quantum open systems , 1992 .

[54]  J. Michl,et al.  High triplet yield from singlet fission in a thin film of 1,3-diphenylisobenzofuran. , 2010, Journal of the American Chemical Society.

[55]  Rudolph A. Marcus,et al.  Chemical and Electrochemical Electron-Transfer Theory , 1964 .

[56]  Timothy C. Berkelbach,et al.  Reduced density matrix hybrid approach: an efficient and accurate method for adiabatic and non-adiabatic quantum dynamics. , 2011, The Journal of chemical physics.

[57]  Manuel Ligges,et al.  The energy barrier in singlet fission can be overcome through coherent coupling and entropic gain. , 2012, Nature chemistry.

[58]  M. Pope,et al.  Effect of Magnetic Field on the Fluorescence of Tetracene Crystals: Exciton Fission , 1969 .

[59]  N. E. Coates,et al.  Efficient Tandem Polymer Solar Cells Fabricated by All-Solution Processing , 2007, Science.

[60]  L. Kaake,et al.  Observing the Multiexciton State in Singlet Fission and Ensuing Ultrafast Multielectron Transfer , 2011, Science.

[61]  M. Nakano,et al.  Theoretical Study of Singlet Fission in Oligorylenes. , 2012, The journal of physical chemistry letters.

[62]  Alessandro Troisi,et al.  Charge transport in semiconductors with multiscale conformational dynamics. , 2009, Physical review letters.

[63]  E. Venuti,et al.  Peierls and Holstein carrier-phonon coupling in crystalline rubrene , 2010 .

[64]  S. Mukamel.  Non-markovian theory of molecular relaxation. I. Vibrational relaxation and dephasing in condensed phases , 1979 .

[65]  R. T. Ross,et al.  Efficiency of hot-carrier solar energy converters , 1982 .

[66]  C. E. Swenberg,et al.  Bimolecular radiationless transitions in crystalline tetracene , 1968 .

[67]  Priya J. Jadhav,et al.  Singlet exciton fission in nanostructured organic solar cells. , 2011, Nano letters.

[68]  Gregory D. Scholes,et al.  Rate expressions for excitation transfer. III. An ab initio study of electronic factors in excitation transfer and exciton resonance interactions , 1995 .

[69]  Timothy C. Berkelbach,et al.  Reduced density matrix hybrid approach: application to electronic energy transfer. , 2011, The Journal of chemical physics.

[70]  Long-time quantum simulation of the primary charge separation in bacterial photosynthesis. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[71]  K. Schulten,et al.  Quest for spatially correlated fluctuations in the FMO light-harvesting complex. , 2011, The journal of physical chemistry. B.

[72]  Lee‐Ping Wang,et al.  The diabatic picture of electron transfer, reaction barriers, and molecular dynamics. , 2010, Annual review of physical chemistry.

[73]  Ariel Biller,et al.  Quasiparticle and optical spectroscopy of the organic semiconductors pentacene and PTCDA from first principles , 2012 .

[74]  T. Holstein,et al.  Studies of polaron motion: Part II. The “small” polaron , 1959 .

[75]  Anthony K. Felts,et al.  The Redfield Equation in Condensed‐Phase Quantum Dynamics , 2007 .

[76]  W. G. Schneider,et al.  Laser Generation of Excitons and Fluorescence in Anthracene Crystals , 1965 .

[77]  Jenny Clark,et al.  Ultrafast dynamics of exciton fission in polycrystalline pentacene. , 2011, Journal of the American Chemical Society.

[78]  A. Nozik,et al.  Multiexciton generation by a single photon in nanocrystals. , 2006, Nano letters.

[79]  A. Troisi,et al.  Hole Migration in DNA: a Theoretical Analysis of the Role of Structural Fluctuations , 2002 .

[80]  S. F. Boys,et al.  Canonical Configurational Interaction Procedure , 1960 .

[81]  Klaus Schulten,et al.  Open Quantum Dynamics Calculations with the Hierarchy Equations of Motion on Parallel Computers. , 2012, Journal of chemical theory and computation.

[82]  Stefan Grimme,et al.  Substantial errors from time-dependent density functional theory for the calculation of excited states of large pi systems. , 2003, Chemphyschem : a European journal of chemical physics and physical chemistry.

[83]  C. Bardeen,et al.  Quantum beats in crystalline tetracene delayed fluorescence due to triplet pair coherences produced by direct singlet fission. , 2012, Journal of the American Chemical Society.

[84]  A. G. Redfield,et al.  The Theory of Relaxation Processes , 1965 .

[85]  R. Merrifield,et al.  Fission of singlet excitons into pairs of triplet excitons in tetracene crystals , 1969 .

[86]  G. Fleming,et al.  On the adequacy of the Redfield equation and related approaches to the study of quantum dynamics in electronic energy transfer. , 2009, The Journal of chemical physics.

[87]  Werner,et al.  Novel optimization principles and efficiency limits for semiconductor solar cells. , 1994, Physical review letters.

[88]  Diana E Schlamadinger,et al.  Triplet excitons of carotenoids formed by singlet fission in a membrane. , 2011, Chemphyschem : a European journal of chemical physics and physical chemistry.

[89]  J. Skinner,et al.  Pure dephasing of a two-level system , 1986 .

[90]  Michael J. Tauber,et al.  High-yield singlet fission in a zeaxanthin aggregate observed by picosecond resonance Raman spectroscopy. , 2010, Journal of the American Chemical Society.

[91]  M. Beck,et al.  The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propa , 1999 .

[92]  Paul M Zimmerman,et al.  Singlet fission in pentacene through multi-exciton quantum states. , 2010, Nature chemistry.

[93]  Michael J. Tauber,et al.  Singlet fission in carotenoid aggregates: insights from transient absorption spectroscopy , 2012, Optics & Photonics - NanoScience + Engineering.

[94]  A. D. Vos,et al.  Detailed balance limit of the efficiency of tandem solar cells , 1980 .

[95]  Hao‐Li Zhang,et al.  Solution processed pentacene thin films and their structural properties , 2007 .

[96]  Masayoshi Nakano,et al.  Diradical Character View of Singlet Fission , 2012 .

[97]  D. Reichman,et al.  Reference system master equation approaches to condensed phase charge transfer processes. I. General formulation , 2001 .

[98]  Angel Rubio,et al.  Electronic structure and excitations in oligoacenes from ab initio calculations. , 2006, The Journal of chemical physics.

[99]  W. Schmidt,et al.  Diels-Alder reactivity of polycyclic aromatic hydrocarbons. 1. Acenes and benzologs , 1980 .

[100]  Eunji Sim,et al.  Path Integral Simulation of Charge Transfer Dynamics in Photosynthetic Reaction Centers , 1997 .

[101]  J. Freed,et al.  A comparison of generalized cumulant and projection operator methods in spin‐relaxation theory , 1975 .

[102]  Mark A Ratner,et al.  Maximizing singlet fission in organic dimers: theoretical investigation of triplet yield in the regime of localized excitation and fast coherent electron transfer. , 2010, The journal of physical chemistry. B.

[103]  Louie,et al.  Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies. , 1986, Physical review. B, Condensed matter.

[104]  Handong Sun,et al.  Singlet fission in rubrene single crystal: direct observation by femtosecond pump-probe spectroscopy. , 2012, Physical chemistry chemical physics : PCCP.

[105]  Matthew T. Whited,et al.  Efficient singlet fission discovered in a disordered acene film. , 2012, Journal of the American Chemical Society.

[106]  L. Reining,et al.  Electronic excitations: density-functional versus many-body Green's-function approaches , 2002 .

[107]  Nancy Makri,et al.  Path integrals for dissipative systems by tensor multiplication. Condensed phase quantum dynamics for arbitrarily long time , 1994 .

[108]  Spencer R Pruitt,et al.  Fragmentation methods: a route to accurate calculations on large systems. , 2012, Chemical reviews.

[109]  D. Coker,et al.  Influence of environment induced correlated fluctuations in electronic coupling on coherent excitation energy transfer dynamics in model photosynthetic systems. , 2012, The Journal of chemical physics.