Microscopic theory of singlet exciton fission. I. General formulation · 2014-05-16 · THE JOURNAL OF CHEMICAL PHYSICS 138, 114102 (2013) Microscopic theory of singlet exciton ﬁssion

Microscopic theory of singlet exciton fission. I. General formulationTimothy C. Berkelbach, Mark S. Hybertsen, and David R. Reichman

Citation: The Journal of Chemical Physics 138, 114102 (2013); doi: 10.1063/1.4794425 View online: http://dx.doi.org/10.1063/1.4794425 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/138/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Triplet diffusion in singlet exciton fission sensitized pentacene solar cells Appl. Phys. Lett. 103, 153302 (2013); 10.1063/1.4824420 Nanoscopic mechanisms of singlet fission in amorphous molecular solid Appl. Phys. Lett. 102, 173301 (2013); 10.1063/1.4795138 Excited state dynamics in solid and monomeric tetracene: The roles of superradiance and exciton fission J. Chem. Phys. 133, 144506 (2010); 10.1063/1.3495764 Exciton relaxation in KBr and CaF 2 at low temperature: molecular dynamics study Low Temp. Phys. 29, 754 (2003); 10.1063/1.1614185 A dynamic model for exciton self-trapping in conjugated polymers. I. Theory J. Chem. Phys. 112, 5399 (2000); 10.1063/1.481109

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THE JOURNAL OF CHEMICAL PHYSICS 138, 114102 (2013)

Microscopic theory of singlet exciton fission. I. General formulationTimothy C. Berkelbach,1,a) Mark S. Hybertsen,2,b) and David R. Reichman1,c)

1Department of Chemistry, Columbia University, 3000 Broadway, New York, New York 10027, USA2Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973-5000, USA

(Received 27 November 2012; accepted 21 February 2013; published online 15 March 2013)

Singlet fission, a spin-allowed energy transfer process generating two triplet excitons from one sin-glet 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, microscopictheory of singlet fission is lacking. In this work, we assemble the components of a comprehensivemicroscopic theory of singlet fission that connects excited state quantum chemistry calculations withfinite-temperature quantum relaxation theory. We elaborate on the distinction between localized di-abatic and delocalized exciton bases for the interpretation of singlet fission experiments in both thetime and frequency domains. We discuss various approximations to the exact density matrix dynam-ics and propose Redfield theory as an ideal compromise between speed and accuracy for the detailedinvestigation of singlet fission in dimers, clusters, and crystals. Investigations of small model sys-tems based on parameters typical of singlet fission demonstrate the numerical accuracy and practicalutility of this approach. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4794425]

I. INTRODUCTION

The Shockley-Queisser limit places the maximal effi-ciency of a single-junction solar cell at about 31%.1 Promis-ing technologies aimed at exceeding this limit include tandemsolar cells,2, 3 hot carrier collection,4–6 and multiple excitongeneration (MEG).7, 8 In MEG and its molecular analogue,singlet fission, a single absorbed photon generates two ormore excitons each of lower energy, eventually yielding twoor more electron-hole pairs. This mechanism results in the-oretical solar cell efficiencies of almost 50%. Singlet fissionis a particularly promising technology in inexpensive organicsolar cells, whose efficiencies to date remain well below thatof their more expensive inorganic counterparts. Proposals toutilize singlet fission in this manner have targeted covalentlylinked dimers for use in dye-sensitized cells9 as well as crys-talline materials for more traditional heterojunction cells.10

Despite initial reports of singlet fission over 40 yearsago,11–14 an explosion of experimental studies have emergedonly recently due to the aforementioned potential for pho-tovoltaic utility. Singlet fission, as typically measured bythe observation of triplets in the form of delayed fluores-cence (DF) or transient absorption (TA), has been found tovary from system to system both in total yield and over-all timescale, making the search for unifying principles verydifficult. The authoritative review by Smith and Michl15

effectively summarizes the state of the field up to 2010.Since then, singlet fission has been further investigated byTA in thin films of diphenylisobenzofuran,16 by DF andTA in crystalline tetracene,17, 18 by time-resolved two-photonphotoemission19 and TA20 in crystalline pentacene, by TAin solution and crystalline rubrene,21 and even by DF and

a)Electronic mail: [email protected])Electronic mail: [email protected])Electronic mail: [email protected].

TA in amorphous films of diphenyl tetracene.22 Singlet fis-sion has also been investigated in carotenoids via resonanceRaman spectroscopy23, 24 and TA.25 Although still far awayfrom commercial use in solar cells, singlet fission has been in-vestigated in pentacene-perylene blend films26 and even suc-cessfully incorporated into heterojunction solar cells utilizingphthalocyanine, tetracene, and C60.10

These many enlightening experiments notwithstanding,the dynamical mechanism of singlet fission is still not wellunderstood. Previous theoretical work has focused almostentirely on identifying the quantum mechanical states in-volved, including their wavefunction character and energeticordering.27–32 Perhaps most significantly, high-level quantumchemistry calculations have identified a multi-exciton statethat is composed of two triplets coupled into an overall spinsinglet.27, 28 The transition to this multi-exciton state is thusspin-allowed and should proceed rapidly, while its triplet-triplet character suggests that it should naturally relax toseparated triplets on a longer timescale. Unfortunately, themulti-exciton nature of this state implicitly prevents its directphotoexcitation such that the state is spectroscopically “dark”and difficult to observe. However, time-resolved two-photonphotoemission,19, 33 transient absorption,17 and delayedfluorescence18 spectroscopic measurements have providedevidence of this multi-exciton state in ultrafast singlet fission.

A simple four-electron four-orbital model suggests atleast two viable mechanisms for the transition from an ini-tially excited intramolecular singlet state, S1, to the multi-exciton triplet-triplet state, TT.15 The first is a mediatedmechanism, whereby a charge transfer state acts as an inter-mediate in the transition from S1 to TT; theoretical studiesof this mechanism in coupled molecular dimers have consid-ered the static electronic parameters,34 as well as the real-timedynamics in the limit of fast coherent transfer35 and in thepresence of a low-frequency solvent bath.36 Alternatively,a direct mechanism has also been implicated, whereby the

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Coulomb potential yields a direct interaction between S1 andTT, avoiding any intermediates. Some authors have invokedsuch a proposal to explain fission in crystalline tetracene andpentacene, based both on experiment19 and quantum chem-istry calculations of clusters.28

An internally consistent theory of singlet fission phenom-ena must comprise a currently nonexistent unification of staticelectronic structure and dynamic relaxation mechanisms. Inthis first article of a series, we pursue this goal, presenting afully microscopic theoretical formalism tailored to the investi-gation of singlet fission in molecular systems. Our goal is theidentification, extension, and marriage of existing techniquesof electronic structure theory and microscopic quantum dy-namics for the efficient and accurate treatment of singlet fis-sion in organic molecules and bulk materials. Such a synthesiselucidates experimental results in both the time and frequencydomains and allows for studies of competing mechanisms aswell as quantitative predictions. Our approach is related inspirit to treatments of excitation energy transfer in photosyn-thetic pigment protein complexes, where reduced density ma-trix simulations similar to those proposed here have enjoyedgreat success in understanding quantum effects in complex,multi-state biological systems.37–39 The second and third ar-ticles of this series will make clear the utility of this formal-ism as we investigate singlet fission in dimers and crystals,respectively.

The layout of this paper is as follows. In Sec. II, wepresent a minimal electronic structure model capable of de-scribing all relevant states and couplings for the problem ofsinglet fission. We then proceed in Sec. III to describe a non-Markovian quantum master equation approach for the de-scription of relaxation mechanisms arising from the couplingof electronic degrees of freedom with nuclear vibrations insystems undergoing singlet fission. We present numerical ex-amples of our approach in Sec. IV as applied to simple modelsystems of singlet fission, benchmarking our results againstnumerically exact calculations. In Sec. V, we reflect on ourapproach and conclude.

II. ELECTRONIC STRUCTURE OF SINGLET FISSION

Our theoretical framework begins with the electronicstructure of singlet fission chromophore systems in the limitof frozen nuclei. In Sec. II A, we will emphasize the utilityof a generically defined diabatic basis of excited states, andin particular how they should be interpreted, thereby provid-ing a rigorous and important language with which to speakabout the electronic structure of singlet fission. As a prac-tical, physically intuitive example of such a framework, inSec. II B, we will present a limited configuration interaction(CI) description of these diabatic excited states. This modelquantum chemical formalism will be recognized as a modestgeneralization of the picture proposed by Smith and Michl15

to understand singlet fission in dimers. Our contributions areto distinguish between the diabatic basis and the exciton basis,as related to both theoretical development and experimentalinterpretation, to formalize this procedure within the contextof ab initio quantum chemistry, and to generalize to the caseof more than two molecules. A higher-level nonorthogonal CI

approach which is very similar in spirit, building up a molec-ular Hamiltonian from diabatic “base states,” has recentlybeen employed for singlet fission by Havenith et al.32 We willbriefly discuss alternative quantum chemistry approaches inSec. II C before assessing the accuracy and summarizing theproposed electronic structure formalism in Sec. II D.

A. The basis of diabatic states

Consider N molecules in an arbitrary geometry. We startby defining a basis of diabatic electronic states, i.e., thosestates whose quantum mechanical character is well-definedand presumed to be independent of the molecular geometry.For an accessible discussion of diabatic states in the contextof electron transfer, see Ref. 40.

The desired basis begins with the exact many-electronwavefunction corresponding to the (singlet) ground stateof the system, �S0 (r1, r2, . . . , rN ) ≡ 〈r|S0〉. All remainingstates in the minimal singlet fission diabatic basis are exci-tations above the ground state, although they are not in gen-eral eigenstates of the electronic Hamiltonian. Specifically,the basis must include all N excited states, which may becharacterized by molecule n being in its first excited singlet(or S1) state, |(S1)n〉, sometimes referred to as Frenkel exci-tations. The next class of excited states are charge-transfer(CT) states, where molecule m has a single positive chargeand molecule n has a single negative charge, denoted |CmAn〉(C for cation and A for anion). The final class of necessaryexcited states comprises those multi-exciton states describedas being a spin-adapted combination of triplet excitations onmolecules m and n, forming an overall singlet, |TmTn〉.

Double excitations that instead couple two singlets arepresumed too high in energy to be relevant for singlet fis-sion and are consequently neglected, as are all double exci-tations involving more than two molecules. States of differ-ing multiplicity (such as triplet- and quintet-coupled triplets,3TT and 5TT) can also be included. While a purely electronicHamiltonian does not couple such states of different multi-plicity, the spin dipole-dipole Hamiltonian does.15 We neglectthis Hamiltonian in our present formalism because it is sig-nificantly weaker, and we instead only focus on the short-time formation of the spin-singlet multi-exciton state, |TmTn〉.However, including spin dipole-dipole and Zeeman interac-tions in the presence of a magnetic field would be impor-tant but straightforward modifications to study such long-timeeffects.

Although these many-electron basis states, which we willdenote generically by |i〉 and |j〉, are not eigenstates of theelectronic Hamiltonian and do not constitute a complete basis,one may consider the projection of the true electronic Hamil-tonian onto this basis,

Hel ≈∑ij

|i〉〈i|Hel |j 〉〈j |. (1)

We wish to emphasize that the states defined above merelyconstitute a physically-motivated (diabatic) basis. Many ex-perimental measurements instead observe the so-called ex-citon basis, which is obtained from the diagonalization ofthe electronic Hamiltonian, Hel , yielding eigenstates that


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are a mixture of the diabatic states. The exciton basis mayalso be considered a proxy for the fully adiabatic basis,which diagonalizes the electronic plus nuclear potential en-ergy Hamiltonian, a duality which becomes exact in the limitof weak system-bath coupling. Therefore, one must exercisegreat caution when speaking about the character of observedstates, e.g., “charge-transfer,” and when discussing results andproposing mechanisms in terms of these states. This distinc-tion must also be kept in mind for traditional excited stateelectronic structure calculations, which inherently probe theadiabatic, and not the diabatic basis.

In light of the above proviso, one may naturally questionthe utility of this diabatic basis. We propose three reasons tobegin the theoretical development of singlet fission from thisbasis:

(i) The physical character of the basis states aids in inter-preting the nature of observed exciton eigenstates, al-lowing for a means to quantify statements such as “amixture of charge-transfer and Frenkel excitations.” Thislatter example will play a prominent role in our futurework on singlet fission in crystals. Similarly, this prin-ciple underlies the coherent superposition approximationrecently proposed to explain MEG in nanocrystals41, 42

and singlet fission in pentacene,19 wherein single- andmulti-exciton (diabatic) states are coupled to yieldan excitonic eigenstate that is a superposition ofthe two.

(ii) The local diabatic basis can yield accurate results, whichcomputationally scale very favorably. Proximity argu-ments alone can naturally suggest coupling terms thatmay be approximated or neglected entirely. Using suchapproximations, to be discussed in more detail in our nextpaper, one may easily build up a large molecular aggre-gate Hamiltonian using only diabatic energies and cou-plings from monomers, dimers, or small clusters, whichmay be computed with very high accuracy. This philoso-phy is reminiscent of fragmentation methods in the pur-suit of linear scaling quantum chemistry.43

(iii) Finally, the molecular character of the diabatic basis al-lows for a straightforward extension to include couplingto molecular vibrations, which naturally separate into in-tramolecular and intermolecular modes, as we detail inSec. III.

Clearly, the accurate construction of diabatic states marksan important research goal for ab initio simulations of sin-glet fission. While our approach here and henceforth em-ploys a constructive strategy, i.e., a direct construction ofdiabats without explicit reference to the adiabatic states ofthe extended system, an alternative route would employ de-ductive strategies that attempt to obtain approximate diabatsgiven a set of adiabats. This latter set of states is more easilyobtained at high accuracy from existing quantum chemistrymethods, although the non-uniqueness of this diabatizationprocedure results in various competing methods with subtledifferences.40, 44, 45 In any case, the framework presented hereis not limited to the CI-type model Hamiltonian outlined be-low, and more accurate diabatic states, as might be obtainedfrom multi-reference quantum chemistry methods,32 can be

naturally incorporated into the dynamical scheme to be dis-cussed in Sec. III.

B. A minimal, truncated CI basis

The accurate quantum mechanical calculation of excitedstates in large molecular systems is still a difficult challenge(see Refs. 27 and 28 for examples of recent high-level quan-tum chemistry calculations as applied to singlet fission) andthus we consider here the simplest possible model Hamilto-nian approach that captures the essential physics contained inthe diabatic framework outlined above. Specifically, we con-sider the minimal active space of all Hartree-Fock (HF), orHF-like, highest occupied and lowest unoccupied molecularorbitals (HOMOs and LUMOs) of the isolated molecules; ex-tension to include additional frontier orbitals is straightfor-ward. We furthermore restrict the electronic structure calcula-tion to all single and select double excitations, the latter ensur-ing treatment of the bi-excitonic triplet-triplet state. If done asa purely ab initio theory, this approach would be somewhatakin to configuration interaction46 with single and (select)double excitations (CISD) with the frozen core and deletedvirtuals approximations, or alternatively a type of (severely)restricted active space CISD. However, our formalism differsslightly in that we consider excitations among the isolatedmolecular orbitals, rather than among the HF orbitals of thefull interacting system.

To make our description more precise, we define the cre-ation (annihilation) operator for the HOMO of molecule nwith spin σ as c

†H,n,σ (cH, n, σ ) and likewise for the LUMO.

The ground state is thus approximately given by

|S0〉 =N∏

n=1

∏σ=↑,↓

c†H,n,σ |0〉, (2)

where |0〉 is the vacuum state of inactive core orbitals, thusfilling the HOMO of all molecules. As discussed above, thisstate is not the result of a self-consistent HF procedure. Fromthis ground state, we will generate the three types of excitedstates described above in Sec. II A. Because the electronicHamiltonian is spin-conserving, we take symmetry-adaptedlinear combinations of select excitations to generate simulta-neous eigenstates of the Sz and S2 operators, with eigenvaluesof 0 for both (sometimes called configuration state functions).

The first type of state is the local Frenkel singlet excita-tion on molecule n, given by

|(S1)n〉 = 1√2

(c†L,n,↑cH,n,↑ + c†L,n,↓cH,n,↓)|S0〉. (3)

In addition to the above intramolecular excitation, the singleexcitations also generate our second type of state, namely theintermolecular charge-transfer excitation obtained by excitingan electron from the HOMO of molecule m to the LUMO ofmolecule n (n = m),

|CmAn〉 = 1√2

(c†L,n,↑cH,m,↑ + c†L,n,↓cH,m,↓)|S0〉, (4)

where C and A denote the cationic and anionic species, re-spectively. The above two types of excited states combine


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to yield all possible single excitations, so that stopping atthis point would constitute a full CI-singles (CIS) within theHOMO-LUMO space.

However, as discussed above, the problem of singlet fis-sion necessarily requires our third type of state, a double ex-citation coupling two intramolecular triplet excitations into astate with overall singlet character,

|TmTn〉 = 1√12

[2c†L,n,↓c

†L,m,↑cH,n,↑cH,m,↓

+ 2c†L,n,↑c

†L,m,↓cH,n,↓cH,m,↑

− c†L,m,↓c

†L,n,↓cH,n,↓cH,m,↓

+ c†L,m,↓c

†L,n,↑cH,n,↑cH,m,↓

+ c†L,m,↑c

†L,n,↓cH,n,↓cH,m,↑

− c†L,m,↑c

†L,n,↑cH,n,↑cH,m,↑]|S0〉. (5)

Finally, we point out that because the molecular orbitalsof distinct isolated molecules are not necessarily orthogonalto one another, the use of creation and annihilation opera-tors acting in the space of these orbitals is not strictly rig-orous. While one could imagine employing suitably orthog-onalized molecular orbitals that retain the localized nature ofisolated orbitals, the actual overlap in molecular dimers andcrystals is often negligibly small, thus justifying the theory inits present form. Should the overlap be retained, Ghiggino andco-workers47, 48 have provided explicit expressions and conve-nient approximations for the related problem of CT-mediatedsinglet and triplet energy transfer; the extension to include thecorrelated triplet pair state would be straightforward.

Having defined a set of diabatic basis states, it thus re-mains to calculate all matrix elements of the electronic Hamil-tonian, 〈i|Hel|j 〉. While the calculation is straightforward, theresults are cumbersome, and so we include the explicit re-sults in Appendix A. As discussed more below in Sec. II D,the diagonal matrix elements (energies) are only expected tobe of qualitative accuracy but can provide useful insight, andlikewise for the off-diagonal elements (couplings). For exam-ple, the couplings naturally separate into two classes: thosecontaining one-electron integrals and those containing onlytwo-electron integrals. The one-electron integrals include thesimple kinetic energy term, describing favorable charge delo-calization, or “hopping.” Such one-electron integrals are ex-pected to be one or more orders of magnitude larger thanthe two-electron ones. Reasonable estimates for typical sin-glet fission chromophores in close proximity are 50–100 meVfor one-electron integrals32, 34, 35, 49 and 5 meV or less for two-electron integrals.32, 49 These simple analytical expressionsand order of magnitude estimates contribute to the interpre-tation of singlet fission in terms of mediated and direct mech-anisms. Qualitatively, the mediated mechanism proceeds viatwo one-electron processes, whereas the direct mechanismproceeds by one two-electron process. Which of these twodiametric mechanisms prevails in a given system of interestwill depend sensitively on the relative energies of the diabaticstates and the dynamics of the nuclear degrees of freedom, aswill be discussed in Sec. III.

C. Aside regarding wavefunction free methods

Although the formalism here has employed the HF or-bitals to construct a many-electron basis, we pause to considersome alternatives. At least at the level of single excitations,many other electronic structure theories can be reduced to aneigenvalue equation for the transition energies, much like CIS.Note that the CIS theory amounts to the diagonalization of aneffective two-particle Hamiltonian,

Hij,kl = δikδjl(εj − εi) + (fi − fj )Kij,kl, (6)

where fi is the ground-state occupancy of orbital i and

Kij,kl = 2(il|jk) − (il|W (r1, r2)|kj ). (7)

Clearly, for CIS, W (r1, r2) = r−112 . Physically, the vertex K

describes the interaction between single-particle excitationsi → j and k → l. If the original single-particle states are not agood approximation to the quasiparticles of the system, as de-termined, e.g., by comparison with electron affinity and ion-ization energies, then the HF excitations are in some sensea poor starting point on which to build interactions. In otherwords, the true many-body excitations will require contribu-tions from many single-particle excitations.

Instead, one could start from a ground state density func-tional theory (DFT) calculation of the isolated molecules,and then consider excitations within the Kohn-Sham (KS)orbitals; we will not dwell here on the physical reasons forwhich the KS orbitals may be better single particle states.Suffice it to say that this approach is adopted in time-dependent DFT (TD-DFT)50–52 and many-body Green’s func-tion approaches,53–55 both of which typically yield resultssuperior to those of HF-based CIS, finding many-body exci-tations strongly dominated by far fewer single-particle excita-tions. For example, the Green’s function based Bethe-Salpeterequation (BSE) in practice adopts the form Eq. (6) with per-turbatively corrected orbital energies and a statically screenedinteraction,

W (r1, r2) ≈∫

d rε−1(r1, r, ω = 0)|r − r2|−1, (8)

where ε(r1, r2, ω) is the frequency-dependent dielectricfunction.54 In the crude limit where ε−1(r1, r2, ω = 0)= ε−1δ(r1 − r2), with ε a dielectric constant, one arrives atsimply static screening of the direct Coulomb term,

Kij,kl = 2(il|jk) − ε−1(il|kj ). (9)

On the other hand, the inclusion of doubly excited statespresents an ongoing challenge to these former methodolo-gies, making them difficult to employ in an internally consis-tent theory of singlet fission. Recent work has demonstratedthat higher excitations only arise in the above theories withthe retention of a frequency dependent interaction kernel.56, 57

Specifically, one solves the eigenvalue-like equation, H(ω)c= ωc, with

Hij,kl(ω) = δikδjl(εj − εi)

+ (fi − fj )[2(il|jk) − (il|W (r1, r2, ω)|kj )].

(10)


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The operator W (ω) is related to the dynamically screened di-electric function in BSE and to the exchange-correlation ker-nel in TD-DFT (i.e., the adiabatic approximation precludesobservation of multiple excitations in TD-DFT). We considerthis a very interesting research focus for singlet fission and asubject of future work.

D. Accuracy and summary

Based on the preceding discussion, the numerical resultsof the approach proposed in Sec. II B will only be of qual-itative accuracy. The diagonal energies, 〈i|Hel |i〉, should beconsidered estimates of their true values, with some leewayfor semi-empirical adjustment. For example, we find that thegas-phase S0 → S1 transition energy of pentacene predictedby the above approach with a 6-31G(d) basis set is 3.76 eV,to be compared to the experimental value of approximately2.3 eV.58, 59 Including the additional dynamical correlationarising from the frozen orbitals (i.e., not just the HOMO andLUMO) yields the improved value of 2.81 eV. While TD-DFTis typically expected to be an improvement, it was shown pre-viously to predict values of 1.64 eV and 1.90 eV, for the PBEand B3LYP functionals, respectively.60 Thus, even purport-edly sophisticated methods yield excitation energies with er-rors ranging from 0.4 to 0.7 eV.60, 61 Interestingly, the ad hocBSE-like prescription, Eq. (9), using only the DFT HOMOand LUMO from B3LYP and the dielectric constant of pen-tacene ε = 3.6, predicts a transition energy of 2.95 eV, muchimproved from the HOMO-LUMO CIS result (note that thereis, however, no a priori reason that the dielectric constantfor bulk pentacene should be physically meaningful for a sin-gle molecule). Only multi-reference perturbation theory27 andfull many-body GW /BSE calculations62 yield quantitative ac-curacy, predicting 2.1 and 2.2 eV, respectively. Similarly, theelectronic couplings in our HOMO-LUMO basis may notbe quantitatively accurate, but have already been shown inother work to provide useful qualitative insight into the effi-ciency of singlet fission through investigation of their relativemagnitudes34 and dependence on molecular orientation.15 Itmay thus be permissible to uniformly scale the electronic cou-pling matrix elements when investigating singlet fission.

To summarize, we argue that the diabatic basis, com-prising states that are easily characterized and energies andcouplings that are straightforwardly calculated, acts as thecrucial conceptual intermediate between high-level quantumchemistry calculations, which inherently yield electronicallyadiabatic states that are difficult to characterize, and micro-scopic quantum master equations, which are required to ac-curately treat thermally induced relaxation effects, the topicof Sec. III.

III. SYSTEM-BATH QUANTUM DYNAMICS

In this section, we consider the coupling of electronic(system) and nuclear (bath) degrees of freedom. Although thetreatment is relatively standard and can be found in textbooks,see, e.g., Ref. 63, we include the derivation in Appendix B,to emphasize the microscopic connection to the diabatic ba-

sis introduced above. The result is the Hamiltonian describedbelow.

A. System-bath Hamiltonian

To include the effects of electron-phonon coupling, weemploy a system-bath type Hamiltonian

Htot = Hel + Hel−ph + Hph (11)

with the electronic Hamiltonian calculated at the ground-stategeometry in terms of the diabatic states described in Sec. II,

Hel =∑

i

|i〉Ei〈i| +∑ij

|i〉Vij 〈j |, (12)

the bilinear electron-phonon coupling,

Hel−ph =∑

i

|i〉〈i|∑

k

ck,i qk +∑ij

|i〉〈j |∑

k

ck,ij qk (13)

with ck, 0 = 0, and the free phonon Hamiltonian,

Hph =∑

k

[p2

k

2+ 1

2ω2

k q2k

]. (14)

In the above, i and j index the diabatic electronic basis states,and k indexes both the inter- and intra-molecular (groundstate) normal modes of the system.

The molecular vibrations and phonons are completely de-scribed by their spectral density,

Jij (ω) = π

2

∑k

c2k,ij

ωk

δ(ω − ωk). (15)

Physically, the spectral density encodes the distribution ofnormal mode frequencies weighted by the strength withwhich each mode couples to the energy level of diabatic statei (Jii(ω)) or to the electronic coupling between states i and j(Jij(ω)). In practice, spectral densities (obtained in a mannerto be described) are usually fit to a numerically convenientfunctional form, J(ω) = λF(ω/), parametrized by the reor-ganization energy, λ = π−1

∫dωJ(ω)/ω and a characteristic

frequency .Atomistically, the spectral densities may be calculated

through a combination of classical molecular mechanics andquantum chemistry calculations. In one approach, a directdiagonalization of the molecular mechanics Hessian yieldsphonon frequencies ωk and displacement vectors, and quan-tum chemistry calculations along these displacements pro-duce the coupling constants ck. Such an approach has beenadopted recently by Girlando et al. in studies of electron andhole transport in rubrene64 and pentacene65 crystals.

Alternatively, by appealing to the quantum-classical cor-respondence of harmonic oscillators, which are presumed tocompose the nuclear bath, one may show that the spectral den-sity can be obtained from the Fourier cosine transform of aclassical correlation function,66, 67

Jij (ω) = ω

kBT

∫ ∞

0Ccl

ij (t) cos(ωt), (16)

where Cclii (t) = 〈δEi(t)δEi(0)〉clT is the energy gap fluctua-

tion correlation function and Ccli =j (t) = 〈δVij (t)δVij (0)〉clT is


128.59.222.12 On: Sat, 12 Apr 2014 18:27:41


the electronic coupling fluctuation correlation function (δX= X − 〈X〉clT ) at temperature T. This latter approach has beenextensively pursued in the present context of organic materi-als by Troisi and co-workers, who have focused on the fluc-tuations and spectral properties of the electronic coupling inDNA,68 pentacene crystals,69 and the discotic phase of hex-abenzocoronene derivatives.70

Given the expense of accurate ab initio quantum chem-istry methods, frequent calculations along the course of amolecular dynamics trajectory are clearly prohibitive. Assuch, it is common to adopt a semi-empirical quantum chem-ical method, such as the spectroscopic parametrization of in-termediate neglect of differential overlap (INDO), which hasan impressive accuracy to cost ratio allowing for the rapidcollection of sufficient statistics. While one could in prin-ciple calculate all the diabatic matrix elements defined inAppendix A, we note that the diagonal elements are domi-nated by the bare orbital energies and the off-diagonal cou-pling matrix elements are dominated by the one-electron cou-pling. Thus, it is reasonable to assume that the stochasticproperties (fluctuation magnitude and timescale) of the fullmatrix element are equivalent to those of its one-electronterms. These latter properties are more commonly evaluatedin the literature, due to their role in the electron and hole trans-port of organic materials.

The last topic of discussion concerns the correlation ofdifferent bath modes, for example, the extent to which thefluctuations of the diabatic energy of state i are correlated withthose of state j. Although there is surely some degree of cor-relation, positive or negative, the effect of its inclusion on thesubsequent dynamics is debatable. In particular, while somestudies have attempted to implicate correlated bath modes inefficient biological energy transport,71, 72 molecular dynamicssimulations of photosynthetic complexes show no significantcorrelations.73, 74 Lacking any firm evidence either way for theproblem of singlet fission, we will let the correlation of differ-ent bath modes be dictated by numerical convenience (usuallypreferring the completely uncorrelated scenario), though it isa topic worthy of further investigation.

B. Reduced density matrix dynamics

The dynamics of the coupled electron system and phononbath is given by the Liouville-von Neumann equation for thetotal density matrix, W (t),

dW (t)

dt= −i[Htot ,W (t)], (17)

the exact solution of which is prohibitively difficult due tothe large Hilbert space associated with the phonon degrees offreedom. However, as long as one is only interested in elec-tronic observables, great simplification occurs when consider-ing the reduced density matrix (RDM) of the system, ρ(t), ob-tained by averaging the total density matrix over the phonondegrees of freedom, i.e., ρ(t) = Trph{W (t)}. The diagonal el-ements of this matrix, ρ ii(t) = 〈i|ρ(t)|i〉, are the populationsof state i and the off-diagonal elements, ρ ij(t) = 〈i|ρ(t)|j〉, arethe coherences between states i and j.

A variety of methods exist for the determination ofthe RDM, each with its own caveats. Although impressiveprogress has been made in the development of numericallyexact methods—including path-integral techniques,75–78 themulti-configurational time-dependent Hartree ansatz,79–81 andhierarchical equations of motion37, 82—we will limit ourselveshere to approximate methods, which are more physicallytransparent and more readily applied to very large systems,as will be demonstrated in our future work on clusters andcrystals.

Approximate methods are generally perturbative in na-ture, and differ in their choice of perturbative parameter.Clearly, the physical problem at hand should dictate the ap-propriate small parameter, thus controlling the accuracy ofthe perturbative approximation. The first common approachis to treat the electronic couplings in the diabatic basis, Vij , tosecond order in perturbation theory, while treating the system-bath interaction exactly; this philosophy comprises Marcus83

and Förster-Dexter84, 85 theories, as well as the more sophis-ticated noninteracting blip approximation.86, 87 Although thismethodology has been previously employed in a study of CT-mediated singlet fission,36 to be discussed later in this paper,we will advocate for an alternative approach, which treats theelectronic couplings exactly in exchange for a perturbativetreatment of the system-bath interaction. The relative meritsof the two approaches will be contrasted in Sec. IV.

Specifically, we shall pursue the use of a Redfield-likeequation,88–92 in either its non-Markovian or Markovian form.Non-Markovian prescriptions can either take a time-local ortime-nonlocal form, which corresponds to a series resumma-tion in terms of different time-ordered cumulants.93–95 Wewill present equations for the time-local form (or partial or-dering prescription), though the treatment of singlet fissiondynamics in terms of the alternative time-nonlocal form (orcomplete ordering prescription) would be straightforward.96

In the excitonic basis of electronic eigenstates, Hel|α〉= ¯ωα|α〉, and adopting the notation ωαβ = ωα − ωβ , thetime-local Redfield equation is given by

dραβ (t)

dt= −iωαβραβ(t) +

∑γ δ

Rαβγ δ(t)ργδ(t), (18)

where the initial condition of the total density matrix im-plicitly takes the factorized form W (0) = ρ(0)e−Hph/kBT /Zph,with the phonon partition function Zph = Trph{e−Hph/kBT }.This initial condition is consistent with an impulsive Franck-Condon excitation at time t = 0. In Eq. (18), the first term onthe right-hand side is responsible for coherent energy trans-fer whereas the second term is responsible for population re-laxation, coherence transfer, and dephasing, and more com-plicated population-to-coherence transfer processes. Explicitexpressions for the Redfield tensor elements, which includeintegrals over thermal bath correlation functions, can be foundin Appendix C.

In the limit where the bath relaxation takes place sig-nificantly faster than that of the electronic system, the time-dependent Redfield equation is well approximated by itsMarkovian form, obtained from Eq. (18) by the replacementRαβγ δ(t) → Rαβγ δ(∞). This approximation clearly simplifies


128.59.222.12 On: Sat, 12 Apr 2014 18:27:41


the form of the density matrix equation and provides a directmicroscopic route to dephasing and relaxation rates, whichare often employed in other contexts as phenomenological pa-rameters. This Markovian approximation should be carefullychecked for its accuracy in each situation of interest.

As is commonly done in theories of exciton transport,one may furthermore employ the secular approximation to theMarkovian Redfield equation, which preserves the positivityof the RDM, i.e., ρ ii > 0.63, 89, 91, 92 The secular approximationamounts to neglecting those elements of the Redfield tensor,Rαβγ δ , for which |ωαβ − ωγδ| = 0. In doing so, one decou-ples the dynamical evolution of populations and coherencesin the excitonic eigenstate basis. In addition to preservingpositivity, the secular approximation furthermore guaranteesthat the system RDM approaches thermal equilibrium at longtimes, i.e.,

ρ(t → ∞) = e−Hel/kBT /Zel, (19)

which is the correct physical result outside regimes of strongsystem-bath coupling.

IV. APPLICABILITY AND ACCURACY OF REDFIELDTHEORY FOR SINGLET FISSION DYNAMICS

Although the presentation of system-bath dynamics upto this point has been largely generic, we now thoroughlydiscuss the applicability of the Redfield equation to the spe-cific problem of singlet fission in organic systems. We mustfirst acknowledge the potential disadvantages of the Redfieldtreatment and the extent to which they affect the reliability ofsuch calculations. The main approximation inherent in thisapproach is the assumption of weak coupling between theelectronic and vibrational degrees of freedom. This couplingcan be quantified approximately by the ratio of the magni-tude of fluctuations in the nuclei, λij, to the frequency of thesefluctuations, ij. For example, in the usual spin-boson modelwith an exponentially cutoff Ohmic spectral density, the di-mensionless Kondo parameter, α = 2λ/¯c, characterizes theoverall strength of the system-bath interaction: at T = 0, thereexists a crossover to completely incoherent dynamics at α

= 1/2 and a localization transition at α = 1.86, 87 Addition-ally, the existing body of numerical work has provided empir-ical guidelines on the applicability of weak-coupling masterequations, see, e.g., Refs. 92 and 97. In summary, if the di-mensionless ratio λij/¯ij is small, then the Redfield approx-imation should be a good one; we furthermore confirm thisresult by comparison with numerically exact results on smallmodel systems in Sec. IV A. Whether this inequality holds ornot for realistic material parameters will depend on the spe-cific system under study.

As a prototypical singlet fission material, consider pen-tacene, which we will study in more detail in the followingpaper. The diagonal reorganization energy and frequencies ofthe electron-phonon coupling have been calculated by quan-tum chemical and molecular dynamics methods65 to be ap-proximately λii = 50 meV and ¯ii = 170 meV, respectively;note that this latter value corresponds to the well-known≈1400 cm−1 aromatic stretching mode. Thus, we see thatthe ratio of the two is indeed significantly smaller than one,

and the Redfield equation should be reasonably accurate. Asa general rule, smaller molecules will undergo larger geom-etry distortions in excited states, i.e., larger λii, and thereforethe Redfield approach may break down.

The Markov approximation to the Redfield equation,as discussed above, relies on timescale separation betweenelectronic and nuclear relaxation, and thus one must com-pare the electronic couplings Vij to the phonon frequen-cies ¯ij. With electronic couplings on the order of 50–100 meV,32, 34, 35, 49 and again vibrational frequencies of170 meV, even the Markov approximation should be reason-ably reliable. The accuracy of the Markov approximation willalso be evaluated empirically in the numerical results to fol-low. In addition to the above mathematical argument, thereis also a more physical implication of the Markov approxi-mation: although the time-dependent variants of the Redfieldequation can describe multi-phonon effects to varying degreesof accuracy, the Markov approximation inherently describesonly single-phonon relaxation mechanisms. This deficiencycan be readily seen in the exciton population relaxation rate,Rααββ , which is proportional to Jij(ωαβ), so that all transitionfrequencies must be matched by a single phonon frequency inthe spectral density.

Potential pitfalls behind us, we now enumerate the manyadvantages of the Redfield formalism. The first obvious ad-vantage, which is shared by a variety of other perturbativemethods, is the clear microscopic formalism. While densitymatrix calculations have been employed for theoretical stud-ies of MEG42 and singlet fission35 as well as for fitting exper-imental singlet fission data,19 such dynamical investigationshave been essentially phenomenological to date. In the ap-proach advocated here, the electronic structure methodologyis directly connected to the molecular structure and micro-scopic relaxation mechanisms. The Redfield tensor prescribestemperature-dependent population relaxation and coherencedephasing rates, which can be traced back to the physicalvibrations of the system under study. When necessary, thetime-dependent Redfield variants even yield non-Markovianbehavior, which, of course, cannot be captured with phe-nomenological time-independent rates.

In addition, like all master equation methods, the Red-field approach scales very favorably in a computational sense.The additional adoption of the Markov and secular approxi-mations further reduces the computational cost. Needless tosay, none of the numerically exact methods alluded to abovetakes on a simple master equation form and thus each hasa significant computational overhead with a scaling that de-pends on the details of the method.

The theoretical study most similar in spirit to our ownis that of Teichen and Eaves36 who sought to quantify the ef-fects of a generally non-Markovian bath of low-frequency sol-vent degrees of freedom and its implications for CT-mediatedsinglet fission. These authors employed methodology simi-lar to the noninteracting blip approximation (NIBA) knownfrom spin-boson theory,86, 87 previously generalized to thecase of multilevel systems76, 98 and recently extended to sit-uations of slow, near-classical bath modes99, 100 in a time-nonlocal formalism. The time-nonlocal methodology, hence-forth referred to as NIBA even for multilevel systems, yields a


128.59.222.12 On: Sat, 12 Apr 2014 18:27:41


non-Markovian master equation for the populations of theRDM in the diabatic basis, Pi(t) = ρ ii(t),

dPi(t)

dt=

∑j

∫ t

0dsKij (t, s)Pj (s), (20)

where

Kij (t, s) = 2|Vij |2 Re⟨exp

(−iH totii t

)exp

(iH tot

jj s)⟩

ph, (21)

and H totii = 〈i|H tot |i〉. Teichen and Eaves36 instead con-

sidered the time-local version of this theory, Pi(t) = Ii(t)+ ∑

j Rij (t)Pj (t), but the two methods should give similarresults, and are identical in the Markovian limit.

As alluded to previously, the NIBA-type master equa-tions are perturbative in the electronic couplings, Vij , andthus the diabatic basis is in some sense a preferred basis. Thenonperturbative effects of strong electronic coupling, yieldingsignificant mixing in the exciton basis, cannot be describedby the NIBA theory. Accordingly, as a theory for popula-tions only, NIBA makes no prediction about coherence vari-ables, ρ ij(t), preventing the transformation to any other elec-tronic basis. As described in more detail in Appendix D, spec-troscopy probes the dynamics of coherences in the excitonbasis, and as such is completely beyond reach of NIBA-basedtheories.

On the contrary, the nonperturbative nature of Redfieldtheory with respect to the electronic Hamiltonian allows foran exact solution of the electronic structure problem in ex-change for an approximate treatment of the system-bath in-teraction. Thus, all questions concerning delocalization, quan-tum coherence, and spectroscopy are readily addressed withthe Redfield framework, as long as the system-bath couplingis not too large. Even in regimes where the time-dependentRedfield theory is pushed past its limits of validity, the secu-lar and Markovian approximations yield a numerically stableLindblad-type master equation, with microscopically-derivedrelaxation and coherence dephasing rates. Interesting recentwork has formulated a stable theory, which reinserts micro-scopic expressions for the population and coherence couplingwithin the Lindblad formalism.101

A. Results for population dynamics

Given the advantages of a Redfield-type approach withrespect to the flexibility of treating populations and coher-ences on equal footing in either the diabatic or exciton bases,as well as the ability to treat extremely large systems, it is nat-ural to ask if such an approach is accurate for typical singletfission systems of current interest. Here, we show with smallmodel systems that indeed treating the system-bath couplingas a perturbative parameter should yield semi-quantitative ac-curacy over a wide range of scenarios rooted physically inthe expected parameter space of acene systems. In all ofthe following results on diabatic population dynamics, wemake comparison with the numerically exact but computa-tionally expensive hierarchical equations of motion (HEOM)methodology,37, 82, 102 as implemented in the Parallel Hierar-chy Integrator (PHI).103 To achieve convergence, we trun-cated the hierarchy at L = 5 and required K = 3 terms in

the Matsubara expansion. We emphasize that although theHEOM formalism is a useful benchmark for small systems,the methodology becomes very expensive to converge forlarger system sizes in the “low-temperature” limit kBT/¯� 1 relevant for aromatic organic molecules.

We begin with a two-state system, which in the con-text of singlet fission may be taken as a model for the di-rect, Coulomb-mediated fission mechanism. The first stateis the photoexcited initial singlet, S1, and the second stateis the multi-exciton configuration, TT. The initial conditionis ρ(0) = |S1〉〈S1| and the dynamics proceeds based on theparameters of the system-bath Hamiltonian defined above.The system-bath coupling will be chosen to take the sim-ple form Hel−ph = ∑

i=S1,T T

∑k,i |i〉ck,i qk,i〈i|, i.e., linear, di-

agonal coupling to uncorrelated bath degrees of freedom.The baths will be characterized by identical Ohmic spectraldensities with a Lorentzian cutoff (sometimes referred to asthe overdamped Brownian oscillator model), Jii(ω) = 2λω/(ω2 + 2).

For concreteness, we will fix the fission to be mildlyexothermic, ES1 − ET T = 75 meV, with a bath cutoff fre-quency ¯ = 150 meV (characteristic of aromatic molecules)and temperature T = 300 K (kBT ≈ 26 meV). However, to in-vestigate the perturbative accuracy of the Redfield and NIBAequations, we will scan the reorganization energy, λ, and elec-tronic coupling, V .

In Fig. 1, we plot the singlet population dynamics for avariety of reorganization energies, with the electronic cou-pling fixed at V = 50 meV. When the system-bath couplingis weak, the quantum beating is dominant and overall relax-ation is slow. It is clear that the timescale of beating shouldnot be confused with the relaxation timescale; only the former

(a) λ = 25 meV

0 50 100 1500.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

nof

S 1

(b) λ = 50 meV

0 50 100 1500.0

0.2

0.4

0.6

0.8

1.0Exact (HEOM)TL RedfieldSM RedfieldNIBA

(c) λ = 100 meV

0 50 100 150

Time t [fs]

0.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

nof

S 1

(d) λ = 200 meV

0 50 100 150

Time t [fs]

0.0

0.2

0.4

0.6

0.8

1.0

FIG. 1. Population dynamics of the two-state singlet fission model describedin the text with ES1 − ET T = 75 meV, V = 50 meV, ¯ = 150 meV (−1

≈ 4 fs), and T = 300 K (kBT ≈ 26 meV), for increasing system-bath cou-pling strength. Approximate results are shown for the time-local (TL) Red-field equation, the secular and Markovian (SM) Redfield equation, and forthe noninteracting blip approximation (NIBA).


128.59.222.12 On: Sat, 12 Apr 2014 18:27:41


is accessible within static electronic structure calculations,whereas the latter requires explicit treatment of the vibrationaldegrees of freedom. As the coupling is increased, all theoriescorrectly predict that the oscillations become damped and therelaxation to TT proceeds more quickly, except in panel (d)where strong system-bath coupling can effectively localizethe initial excitation yielding a decreased rate. We see thatas the system-bath interaction becomes large, the time-localRedfield result becomes inaccurate at long times, even leadingto unphysical negative populations. However, to some extent,the secular and Markov approximations to the Redfield equa-tion prevent such a catastrophe, leading to much more reason-able equilibrium populations. The non-Markovian behavior,on the other hand, can be observed in the short-time dynamics,which are always correctly described by the time-local Red-field equation, but not by its secular, Markovian counterpart.In contrast to the breakdown behavior of the Redfield equa-tion, the NIBA dynamics retain their relative accuracy at allvalues of the system-bath coupling, including the decreasedrate observed in panel (d). This result is to be expected in asmuch as the NIBA theory treats the system-bath interactionexactly. Rather, the NIBA theory is perturbative in the elec-tronic coupling, which is unchanged in all panels of Fig. 1.However, NIBA can be seen to consistently underestimate theequilibrium population of S1. This tendency towards extremelocalization in biased systems is a known deficiency of meth-ods that are perturbative in the electronic coupling.86, 87, 104

The rapid singlet fission observed in Fig. 1, with 100 fstimescales, is due to the rather large value of the direct cou-pling matrix element, V . As alluded to previously, this num-ber is likely significantly smaller than 50 meV and so weshow, in Fig. 2, the effect of reducing the electronic couplingat fixed λ = 25 meV. As V gets progressively smaller, thetimescale of relaxation grows significantly, reaching approx-imately 100 ps for V = 1 meV. At the smallest values of V ,

(a) V = 1 meV

0 40 80 120 160 200

0.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

nof

S 1

(b) V = 5 meV

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0Exact (HEOM)TL RedfieldSM RedfieldNIBA

(c) V = 25 meV

0.0 0.1 0.2 0.3 0.4 0.5

Time t [ps]

0.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

nof

S 1

(d) V = 50 meV

0.00 0.05 0.10 0.15 0.20 0.25

Time t [ps]

0.0

0.2

0.4

0.6

0.8

1.0

FIG. 2. The same as in Fig. 1, but with λ = 25 meV and scanning the elec-tronic coupling, V . Note the changing scale of the time axis.

panels (a) and (b), the relaxation rate can be seen to followthe expected k ∝ V 2 golden rule. Consistent with their per-turbative origins, the NIBA dynamics become quantitativelyexact for vanishing V , whereas Redfield theory’s qualitativeaccuracy is maintained throughout all panels. For this valueof the reorganization energy, the secular and Markov approx-imations to the time-local Redfield equation yield impressivequantitative accuracy for all values of V . This behavior can berationalized based on the previous discussion because evenin panel (d), V � ¯, and so the population dynamics areslower than the bath dynamics. For significantly larger valuesof V , the Markov approximation may break down.105

As another important numerical test, we now considerthe effect of a third state on the dynamics of singlet fission,where an initial state couples to a second state, which in turncouples to a third. This configuration is clearly akin to themediated mechanism, with the three states S1, CT, and TT.Interestingly, the quantum dynamics of such mediating sys-tems has precedent in the donor-bridge-acceptor complexes ofphotosynthetic charge transfer. Over 15 years ago, Makri andco-workers performed numerically exact path integral simula-tions of a three-state model very similar to the one consideredhere and detailed the dynamical features of two previouslyproposed, but physically distinct transport mechanisms.106, 107

The first mechanism is a sequential one, whereby the interme-diate state becomes transiently populated in a scheme well-described by two-step kinetics. This mechanism dominates inenergetic regimes satisfying E1 > E2 > E3. A second mecha-nism, evincing the superexchange phenomenon, employs vir-tual states of the intermediate, which is never directly popu-lated, yielding overall single-step kinetics for the 1 → 3 trans-fer. Superexchange applies when the intermediate state energyis much higher than the other two, E2 � E1 > E3.

In light of the similarity with mediated singlet fission(both qualitatively and quantitatively, see below), we considerit an essential criterion that any adopted quantum dynamicsmethodology be able to capture this effect. To make connec-tion with the singlet fission problem, we will henceforth con-sider the definite state labeling referred to above, namely S1,CT, and TT. For comparison, we adopt the same electronic pa-rameters used in Ref. 106 and the same system-bath couplingused above, Hel−ph = ∑

i=S1,CT ,T T |i〉∑k,i ck,i qk,i〈i|. Again

the baths have an Ohmic spectral density with Lorentzian cut-off parametrized by λ = 25 meV, ¯ = 150 meV, and T= 300 K. We wish to strongly emphasize that although theHamiltonian (electronic parameters to follow) was originallyparametrized based on photosynthetic protein data, the mag-nitude of the parameters is almost identical with those ex-pected of singlet fission.

First, we consider the sequential regime, for which theelectronic Hamiltonian has ES1 = 250, ECT = 200 meV, andETT = 0 meV, with VS1,CT = 3 meV and VCT,T T = 17 meV. InFig. 3(a), we plot the population dynamics of the three statesand find that both Redfield theory and NIBA agree quantita-tively with each other and with numerically exact HEOM dy-namics. Clearly, in this sequential mechanism, S1 populatesCT, which in turn populates TT.

We now turn to the superexchange regime, for whichES1 = 80, ECT = 330 meV, and ETT = 0 meV, with VS1,CT


128.59.222.12 On: Sat, 12 Apr 2014 18:27:41


(a) Sequential

S1

CT

TT

0 2 4 6 8 10 12 14

Time t [ps]

0.0

0.2

0.4

0.6

0.8

1.0Po

pula

tion

ofS 1

,CT

,TT

(b) Superexchange

S1

CT

TT

0 2 4 6 8 10 12 14

Time t [ps]

0.0

0.2

0.4

0.6

0.8

1.0

ExactRedfieldNIBA

FIG. 3. Population dynamics of the three-state model described in the text.While both Redfield theory and NIBA provide a reliable description of dy-namics in the two-step sequential regime, NIBA is qualitatively unable todescribe the superexchange regime. Redfield equation dynamics employ thesecular and Markov approximations and exact results are calculated with theHEOM approach.

= VCT,T T = 30 meV; note that the immense barrier, ECT

− ES1 = 250 meV ≈ 10kBT completely prohibits thermal ac-tivation. Turning to the results in Fig. 3, the Redfield andNIBA dynamics strongly differ, and only Redfield theorygives results in good quantitative agreement with the exactdynamics. In the superexchange limit, the intermediate CTstate is never directly populated, and the kinetics follows asimple S1 → TT dynamical scheme. Thus, in spite of therelatively small electronic coupling values, VS1,CT = VCT,T T

= 30 meV, superexchange must be understood as a higher-order effect. The effective electronic coupling from S1 to TTdue to CT-mixing is Veff ≈ VS1,CT VCT,T T /(ECT − ET T ). Ananalogous expression has been obtained previously in the con-text of CT-mediated singlet and triplet energy transfer.47, 48, 108

Performing second-order perturbation theory with this effec-tive electronic coupling thus gives a rate, which is overallfourth-order, explaining why superexchange eludes the usualsecond-order treatment, such as that employed in NIBA. Red-field theory, on the other hand, is completely nonperturba-tive in the electronic couplings, and is thus entirely capableof capturing this highly relevant phenomenon. Although wewill have more to say about it in our future work, this simplemodel clearly refutes arguments that high-lying CT interme-diates preclude efficient mediated singlet fission.

B. Results for spectroscopy

Finally, we apply the Redfield formalism to the calcu-lation of linear absorption spectroscopy, the formalism ofwhich is described in Appendix D. In this situation, the non-Markovian time-local variant is to be preferred, as it ex-actly solves the so-called pure-dephasing problem appropriatefor the single-molecule absorption spectrum. Employing themethodology described there, we have calculated the absorp-tion of a pentacene molecule in solution, which compares veryfavorably with the experimental spectrum, see Fig. 4. In par-ticular, the phonon sidebands are accurately reproduced, eventhough this is a purely non-Markovian, multi-phonon signa-

2.0 2.2 2.4 2.6

hω [eV]

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

[a.u

.]

TL RedfieldExperimental

FIG. 4. Calculated and experimental absorption spectrum of a single pen-tacene molecule at T = 100 K in solution. The calculation parameters areE(S1) = 2.3 eV, ¯ = 170 meV, and λ = 120 meV (S = 0.7). The bath ismodeled by a single phonon frequency and a homogeneous broadening of30 meV has been applied. Experimental spectrum is from Ref. 109.

ture. As such, this feature cannot be described by the Marko-vian Redfield theory. Comparisons such as this one provideessential benchmarks for the accurate parametrization of bothelectron and phonon degrees of freedom, as well as the inter-action between them.

In the presence of intermolecular interactions, the absorp-tion lineshape will be changed from that of Fig. 4 due to theelectronic coupling to other excited states. To discuss theseeffects, let us introduce the dipole operator which, to a rea-sonable approximation, is given by

μ =∑m

|(S1)m〉μS1〈S0| + H.c., (22)

where H.c. denotes the Hermitian conjugate. This approxi-mation follows from the observation that the transition dipolematrix element for a single excitation from molecule m tomolecule i,

μim ≡ −e

∑n

〈0|rn|�im〉 = −e (Hm|r|Li) (23)

is significantly larger for m = i (Frenkel excitations) than form = i (CT excitations), due to spatial locality. Thus, onlythe intramolecular Frenkel excitation has a non-negligibletransition dipole moment, and charge-transfer and multiple-exciton states do not absorb light. However, the expressionfor the absorption signal (see Eq. (D1)) requires the exci-tonic transition dipole moments, which follow from the aboveequation and the diagonalizing transformation to the excitonbasis,|α〉 = ∑

i Cαi |i〉, as

μα = 〈0|μ|α〉 ≈∑(S1)n

Cα(S1)nμS1 . (24)

It is crucially important to recognize that the extent to whichthe exciton state α is composed of diabatic intramolecularexcitations (Cα

(S1)n) determines the strength with which it ab-

sorbs; this is the phenomenon of intensity borrowing. There-fore, signatures of “dark” diabatic states, such as charge-transfer or multi-exciton states, will appear in absorptionspectra if these states are strongly coupled to the “bright” in-tramolecular Frenkel excitations.


128.59.222.12 On: Sat, 12 Apr 2014 18:27:41


This phenomenon has been previously addressed in theMEG literature within the context of the coherent superposi-tion approximation, wherein the authors concluded that cou-pling to multi-exciton states does not affect the total absorp-tion coefficient, α.42 While we agree that the integrated ab-sorption coefficient is indeed only determined by the brightsinglet excitons,

α =∫ ∞

0dωI (ω) ≈

∑α

|μα|2∫ ∞

0dωδ(ω − ωα0)

=∑

α

|μα|2 =∑

α

∑(S1)m

∑(S1)n

[Cα

(S1)m

]∗Cα

(S1)n |μS1 |2

=∑(S1)m

∑(S1)n

〈(S1)m|(S1)n〉|μS1 |2 = N |μS1 |2, (25)

we argue that the spectral structure (peak positions and in-tensities) is surely affected by the coupling to multi-excitonstates. For example, consider an artificial system composedof the ground state, |0〉, a single bright state, |S1〉, and a dark(multi-exciton) state, |D〉, with equal energies E(S1) = E(D)≡ E and mutual coupling 〈S1|Hel|D〉 ≡ V . The dipole op-erator is simply μ = μ|S1〉〈0| + H.c. In the uncoupled limitV = 0, the electronic Hamiltonian is already diagonal, andEq. (D2) gives the absorption lineshape as I(ω) = |μ|2δ(ω− E), and the absorption coefficient α = |μ|2. However, forV = 0, the excitonic eigenstates are the symmetric and anti-symmetric linear combinations, |α〉 = (2)−1/2(|S1〉 + |D〉), |β〉= (2)−1/2(|S1〉 − |D〉), with energies Eα = E − V , Eβ = E

+ V . In this case, the absorption lineshape shows two weakerpeaks,

I (ω) = 1

2|μ|2 [δ(ω − E + V ) + δ(ω − E − V )] , (26)

but the same total absorption coefficient, α = |μ|2. In light ofthe preceding analysis, we propose that evidence for couplingto dark, perhaps multi-exciton, states should be observable inthe linear absorption spectrum, perhaps contrary to standardintuition.

As a related point, we recall that there have been pro-posals to experimentally seek out real-time quantum beatingas evidence of coupling to multi-exciton states,42 even if, inthe limit of strong coupling, the frequency of beating be-comes too high for experimental resolution in the time do-main. However, the origin of spectral peaks discussed aboveis identical to that of quantum beating, namely the oscilla-tion of quantum coherences at frequencies given by energydifferences. More importantly, in the frequency domain, thepeak splitting is proportional to the strength of the couplingand thus more easily observed for strong coupling. In thissense, linear absorption and real-time quantum beating shouldbe seen as complementary tools for the investigation of cou-pling to multi-exciton states: weak coupling yields negligiblysplit peaks that may be difficult to resolve, but produces slowoscillations in real time that should be easy to observe. Thesituation is reversed for strong coupling, where spectral mea-surements should be preferred. As a proviso, the real-time ob-servation of quantum beating may be an artifact of the diabaticbasis. Specifically, the exciton basis populations may show

pure exponential relaxation, but because the transformationback to the diabatic basis mixes populations with oscillatorycoherences, the diabatic populations appear to exhibit quan-tum beating. Thus, the real-time detection of such beating inpart depends on the basis which is probed experimentally.

The examples discussed in this section on spectroscopyand the preceding one on population dynamics illustrate theutility of a Redfield approach to the description of organicsinglet fission systems. In particular, most materials of cur-rent interest for singlet fission lie in a regime where the ra-tio λii/¯ii < 1, largely due to the dominant coupling tohigh-frequency aromatic carbon bond stretching. For the samereason, these systems generically have bath relaxation timesthat make the simplifying Markov approximation a sensibleone. Finally, the Redfield theory and its variants are non-perturbative in the electronic states. Thus, they do not alterthe underlying description of the frozen electronic structuretheory and are capable of treating higher-order effects such assuperexchange. This last point will be significant in our dis-cussion of singlet fission in pentacene dimers.

V. CONCLUSIONS

To summarize, we have presented a self-contained micro-scopic theory of multi-exciton formation in the context of sin-glet fission. Our formalism emphasizes the difference in elec-tronic bases, diabatic and excitonic (or adiabatic), as appliedto both theoretical development and experimental interpreta-tion. Building on this electronic foundation, we have appliedtechniques from the theory of open quantum systems to de-scribe the finite-temperature quantum dynamics of charge andenergy transfer processes taking place in singlet fission mate-rials. Specifically, such processes are facilitated by phononabsorption and emission, which can be given a microscopicorigin in terms of certain vibrational motions of molecules.

We furthermore discussed various approximate quantummaster equations for the reduced density matrix and foundthat while NIBA-like theories, which are perturbative in theelectronic couplings yield accurate dynamics for two-statesystems in regimes expected to hold in organic systems under-going singlet fission, their perturbative nature is exposed byhigher-order processes in multistate systems, such as the su-perexchange mechanism. On the contrary, Redfield-like the-ories, perturbative in the system-bath coupling, yield reason-ably accurate results in essentially all regimes of relevance forsinglet fission. We additionally elucidated the importance of atheory for both the populations and coherences of the reduceddensity matrix, allowing for the investigation of dynamics indifferent electronic bases as well as prediction of linear spec-troscopies such as absorption. While more accurate quantumdynamics scheme are certainly worthy of consideration, weemphasize the efficiency of Redfield theory, which allows fora rapid, but thorough, investigation of parameter space in bothsmall and large systems, which is ideal for a computationalscreening of efficient singlet fission target materials.

In the small model systems considered here, we havefound that both direct and mediated mechanisms are plau-sible pathways to efficient singlet fission. For reasonableelectronic Hamiltonian parameters, the phonon degrees of


128.59.222.12 On: Sat, 12 Apr 2014 18:27:41


freedom facilitate fission on picosecond timescales. In par-ticular, we drew a potentially useful comparison with chargetransport in photosynthetic donor-bridge-acceptor systems inthe context of CT-mediated fission, wherein both sequentialand superexchange mechanisms should be considered pos-sible. We emphasize that our proposal for a unified frame-work for the microscopic treatment of singlet fission in or-ganic systems is based on accuracy, efficiency, and physicaltransparency. In particular, we have generalized existing tech-niques and used physical arguments and numerical bench-marks to marry them for the purpose of a microscopic and ac-curate treatment of fission in systems ranging from dimers tocrystals. The companion paper to this one begins this programin pentacene dimers while future work will consider large ag-gregates and bulk crystals.

ACKNOWLEDGMENTS

This work was supported in part by the Center forRe-Defining Photovoltaic Efficiency through Molecule ScaleControl, an Energy Frontier Research Center funded by theU.S. Department of Energy, Office of Science, Office of Ba-sic Energy Sciences (Award No. DE-SC0001085). This workwas carried out in part at the Center for Functional Nanomate-rials, Brookhaven National Laboratory, which is supported bythe U.S. Department of Energy, Office of Basic Energy Sci-ences (Contract No. DE-AC02-98CH10886) (M.S.H). T.C.B.was supported in part by the Department of Energy Office ofScience Graduate Fellowship Program (DOE SCGF), madepossible in part by the American Recovery and ReinvestmentAct of 2009, administered by ORISE-ORAU (Contract No.DE-AC05-06OR23100).

APPENDIX A: ENERGIES AND COUPLINGS INTRUNCATED CI BASIS

Using the minimal basis presented in Sec. II B, here wegive formulas for the diagonal and off-diagonal matrix ele-ments of the exact electronic Hamiltonian operator,

Hel =∑ij,σ

hij c†i,σ cj,σ + 1

2

∑ijkl,σσ ′

Vijklc†i,σ c

†j,σ ′cl,σ ′ck,σ ,

(A1)where the sums are over all molecular orbitals of the isolatedmolecules (the indices i, j, k, l now contain both the moleculeand its orbital) and

hij =∫

d3rφ∗i (r)

[−1

2∇2

r + Vel−nuc(r)

]φj (r) ≡ (i|h|j ),

(A2)

Vijkl =∫

d3r1

∫d3r2φ

∗i (r1)φ∗

j (r2)r−112 φk(r1)φl(r2) ≡ (ij |kl)

(A3)are one- and two-electron integrals over spatial orbitals. Hereand henceforth, we employ atomic units.

The diagonal matrix elements of the Hamiltonian opera-tor in the diabatic basis (“energies”) are given by expressions

available from known configuration interaction theory,

E[S0] =∑i∈S0

2(i|h|i) +∑

i,j∈S0

2(ij |ij ) − (ij |ji), (A4)

E [(S1)n] = E[S0] + Eg + 2KHnLn− JHnLn

, (A5)

E [CmAn] = E[S0] + Eg + 2KHmLn− JHmLn

, (A6)

and

E [TmTn] = E[S0] + 2Eg + JHmHn+ JLmLn

− JLmHm− JLmHn

− JLnHm− JLnHn

+ (1/2)(KHmHn+ KLmLn

+ 3KLnHm+ 3KLmHn

).(A7)

We have introduced the gap Eg = εLm− εHm

and notation fordirect Coulomb integrals, Jij = (ij|ij), and exchange integrals,Kij = (ij|ji).

While one could in principle calculate all possible cou-plings between the previously introduced diabatic states,those couplings involving three or four molecules will be neg-ligibly small due to the weak overlap of the localized molec-ular orbitals. Specifically, we propose to neglect all three-center and higher two-electron integrals. This semi-empiricalapproximation should not drastically affect the results.

Introducing the spatial orbital matrix elements of theFock operator, F ,

(i|F |j ) = (i|h|j ) +∑k∈S0

2(ik|jk) − (ij |kk), (A8)

the remaining off-diagonal matrix elements (“couplings”) canbe evaluated to give

〈CmAn|Hel|(S1)m〉 = (Lm|F |Ln)

+ 2(HmLm|LnHm) − (HmLm|HmLn),

(A9)

〈CmAn|Hel|(S1)n〉 = −(Hm|F |Hn)

+ 2(HnLn|LnHm) − (HnLn|HmLn),(A10)

〈CmAn|Hel|CnAm〉 = 2(HmLm|LnHn) − (HmLm|HnLn),(A11)

〈CmAn|Hel|TmTn〉 =√

3/2{(Lm|F |Hn)

+ (LmLn|HnLn) − (LmHm|HnHm)},(A12)

〈(S1)m|Hel|(S1)n〉 = 2(HmLn|LmHn) − (HmLn|HnLm),(A13)

〈(S1)m|Hel|TmTn〉 =√

3/2{(LmLn|HnLm)

− (HmHn|LnHm)}. (A14)


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We have neglected the coupling to the ground state, e.g.,〈CmAn|Hel|S0〉. Note that such terms do not strictly vanishas they do in traditional CIS (Brillouin’s theorem) becausethe reference state is not the Hartee-Fock solution of the fullmolecular cluster. Nonetheless, because the energy gap be-tween ground and excited states is large, the renormalizationof excited states due to coupling to the ground state should benegligible. Furthermore, although these terms could in prin-ciple facilitate non-radiative decay to the ground state, we as-sume a bottleneck for phonon emission prevents such events,justifying our neglect of such couplings.

APPENDIX B: DERIVATION OF THE SYSTEM-BATHHAMILTONIAN

We begin by considering the nuclear dependence of thediabatic electronic state energies and couplings introducedabove, i.e., Ui( Q) = 〈i|U ( Q)|i〉 and Vij ( Q) = 〈i|U ( Q)|j 〉.To simplify notation, we will employ the Roman charactersi and j to denote diabatic states and Greek characters (α, β,γ , . . . ) to denote the exciton eigenstates.

In the limit where the ground diabatic state is a localminimum, we may perform a second-order Taylor expan-sion of the potential in terms of the mass-weighted coor-dinates Qn ≡ √

Mn( Qn − Q(0)n ) and employ a transforma-

tion to the normal modes that diagonalize the Hessian qk

= ∑Natomn=1

∑3x=1 uk

n,xQn,x , yielding

U0({qn}) ≈ E0 + 1

2

∑k

ω2kq

2k , (B1)

where E0 = U0({0}) and {ω2k} are the 3Natom eigenvalues of

the Hessian.63 By promoting the nuclear coordinates to oper-ators and re-inserting the kinetic energy operator, we arriveat the ground state diagonal matrix element of an electron-phonon Hamiltonian in normal-mode coordinates,

〈0|Htot |0〉 = E0 +∑

k

[p2

k

2+ 1

2ω2

k q2k

]. (B2)

The matrix elements of the total Hamiltonian in the higher-lying diabatic states thus additionally acquire a linear termdescribing the shift of the excited state potential energy sur-face minimum located at {q(i)

k },

〈i|Htot |i〉 = Ui({q(i)k }) +

∑k

[p2

k

2+ 1

2ω2

k

(qk − q

(i)k

)2]

≡ Ei +∑

k

[p2

k

2+ 1

2ω2

k q2k + ck,i qk

], (B3)

where the vertical energy is Ei ≡ Ui({0}) = Ui({q(i)k }) + λii ,

the Holstein-like coupling constants are given by65, 110 ck,i

= −ω2kq

(i)k and the reorganization energy63, 91 of state i is de-

fined as

λii = 1

2

∑k

ω2k

[q

(i)k

]2 = 1

2

∑k

c2k,i

ω2k

. (B4)

It will be convenient to now define the so-called spectral den-sity of the phonons, which completely characterizes the har-

monic environment intrinsic in the normal-mode decomposi-tion. The spectral density of state i is defined by

Jii(ω) = π

2

∑k

c2k,i

ωk

δ(ω − ωk), (B5)

from which the reorganization energy can be rewritten asλii = π−1

∫ ∞0 dωJii(ω)/ω.

The coordinate dependence of the off-diagonal matrix el-ements in these normal modes can then also be evaluated tofirst order,

〈i|Htot |j 〉 = Vij +∑

k

ck,ij qk, (B6)

with the Peierls-like coupling constants given by65, 111

ck,ij = ∂Vij ({qn})∂qk

∣∣∣∣{qn}={0}

. (B7)

These coupling constants can also be incorporated into an off-diagonal spectral density

Jij (ω) = π

2

∑k

c2k,ij

ωk

δ(ω − ωk) (B8)

with a corresponding “reorganization” energy λij = π−1∫ ∞0 dωJij (ω)/ω.

Combining all of the above, we thus arrive at the desiredsystem-bath-type Hamiltonian, Eq. (11).

APPENDIX C: EXPRESSIONS FOR THE REDFIELDTENSOR ELEMENTS

The Redfield tensor is given explicitly as91, 92

Rαβγ δ(t) = �+δβαγ (t) + �−

δβαγ (t)

− δδβ

∑κ

�+ακκγ (t) − δαγ

∑κ

�−δκκβ(t), (C1)

where

�+αβγ δ(t) =

∫ t

0dse−iωγ δs

⟨H

el−ph

αβ (s)H el−ph

γ δ (0)⟩ph

, (C2)

�−αβγ δ(t) =

∫ t

0dse−iωαβ s

⟨H

el−ph

αβ (0)H el−ph

γ δ (s)⟩ph

(C3)

are integrals of thermal bath correlation functions. We haveintroduced the notation

Hel−ph

αβ (t) ≡∑i,j

〈α|i〉eiHpht/¯〈i|Hel−ph|j 〉e−iHpht/¯〈j |β〉

(C4)and 〈. . . 〉ph ≡ Trph{. . . e−Hph/kBT /Zph}. The calculation ofthe thermal bath correlation functions required for the Red-field relaxation tensor is straightforward for the harmonicbaths derived above. Assuming uncorrelated bath fluctua-tions, one finds∑

k,k′ck,ick′,j 〈qk,i(t)qk′,j (0)〉ph = δijCii(t), (C5)

∑k,k′

ck,ij ck′,mn〈qk,ij (t)qk′,mn(0)〉ph = (δimδjn + δinδjn)Cij (t),

(C6)


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and 〈qk,ij (t)qk′,m(0)〉ph = 0. The functions Cij(t) are given bythe usual weak-coupling expressions

Cij (t) = 1

π

∫ ∞

0dωJij (ω)

{coth

(βω

2

)cos(ωt) − i sin(ωt)

}.

(C7)

Observe that in the Markovian limit, Eqs. (C2) and (C3) re-duce to ordinary one-sided Fourier transforms.

APPENDIX D: CALCULATION OF SPECTROSCOPICOBSERVABLES

Spectroscopic observables, such as absorption and emis-sion lineshapes, can be straightforwardly calculated fromRDM dynamics. For example, the ε-polarized absorptionlineshape, Iε(ω), is known to be given by the Fourier trans-form of the dipole-dipole correlation function, Dε(t).63, 112

Within certain approximations,63 the dipole-dipole correla-tion function can be written as Dε(t) = ∑

α|μα, ε |2ρα0(t) withρα0(0) = 1, where μα, ε is the ε component of the transitiondipole moment from the ground state to the electronic excitonstate α, yielding

Iε(ω) =∑

α

|μα,ε |2 Re∫ ∞

0dteiωtρα0(t). (D1)

As a simple example, consider neglecting the electron-phononcoupling, such that ρα0(t) = exp (−iωα0t); then the absorptionspectrum takes the familiar form

Iε(ω) =∑

α

|μα,ε |2 Re∫ ∞

0dteiωt e−iωα0t

=∑

α

|μα,ε |2 δ(ω − ωα0), (D2)

which clearly neglects any broadening or multiphonon ef-fects. In the general case, i.e., by propagating the reduceddensity matrix ρ(0) = |α〉〈0| under the Redfield equation,such environmental effects may be included, as was demon-strated numerically in Sec. IV. Renger and Marcus112 pro-posed the analytically useful procedure whereby one keepsthe full time dependence of the diagonal coherence dephasingtensor, �+

αααα(t), and performs the Markov approximation forthe off-diagonal tensors, �+

αββα , all within the secular approx-imation, i.e.,

dρα0(t)

dt= −iωα0ρα0(t) + Rα0α0(t)ρα0(t) (D3)

with

Rα0α0(t) ≈ −�+αααα(t) −

∑β

�+αββα(t → ∞). (D4)

Before concluding, we consider a specific example of aspectroscopic calculation within the time-dependent Redfieldformalism, relevant to the absorption of a single molecule, theso-called pure dephasing problem.113, 114 In this case, there isno electronic coupling, Vij = 0, and the Hamiltonian is com-

pletely specified by the two level system,

Htot = |S0〉E(S0)〈S0| + |S1〉(

E(S1) +∑

k

ckqk

)〈S1|

+∑

k

(p2

k

2+ 1

2ω2

k q2k

). (D5)

The time-local Redfield equation of motion for the coherenceρS1S0 (t) is simply

dρS1S0 (t)

dt= −i[E(S1) − E(S0)]ρS1S0 (t)

+RS1S0S1S0 (t)ρS1S0 (t), (D6)

which can be straightforwardly solved. With the initial condi-tion ρ(0) = |S1〉〈S0|, one finds

ρS1S0 (t) = exp[−i(ωS1S0 − λS1 )t + g(t) − g(0)], (D7)

where the lineshape function g(t) is given by

g(t) = 1

π

∫ ∞

0dω

JS1 (ω)

ω2

{coth

(βω

2

)cos(ωt) − i sin(ωt)

}.

(D8)

In fact, the pure dephasing problem can be straightfor-wardly solved exactly,114 using the fact that the Hamiltonianis already diagonal in the electronic states and well-knownthermal properties of harmonic oscillators. If one carries outthis procedure, it is found to give exactly the same result asEq. (D7), a remarkable property of the time-local Redfieldequation. Thus, the cumulant resummation inherent in thetime-local formalism is exact for the pure dephasing problem.This result does not hold for the the time-nonlocal approach.

To make our example more specific, consider a single vi-brational mode at frequency , J(ω) = πS2δ(ω − ), suchthat

g(t) = S[coth(β/2) cos(t) − i sin(t)]. (D9)

We have introduced the dimensionless Huang-Rhys factor,S = π−1

∫ ∞0 dωJ (ω)/ω2 making clear that g(0) is, at zero

temperature, identical to the Huang-Rhys factor. Althoughan analytical evaluation of the required Fourier integral,Eq. (D1), is still difficult, it can be straightforwardly cal-culated numerically. We show in Fig. 5 example absorption

(a) S = 0.5

0 1 2 3 4 5

(ω − ωS1S0 + λS1) /Ω

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

[a.u

.]

(b) S = 2.0

0 1 2 3 4 5

(ω − ωS1S0 + λS1) /Ω

0.0

0.2

0.4

0.6

0.8

1.0

FIG. 5. Single molecule absorption spectra at T = 0 for a single vibrationalfrequency, , with the Huang-Rhys parameter, S, as given. Spectra have beenartificially broadened for clarity and normalized to the value of the S = 0.5zero-phonon (0–0) line.


128.59.222.12 On: Sat, 12 Apr 2014 18:27:41


spectra calculated for two different values of the Huang-Rhysparameter, S. Clearly, the well-known vibronic progression isperfectly captured, even in regimes of very strong system-bath coupling. Again we emphasize that this nonperturba-tive multi-phonon behavior is a purely non-Markovian effect,which is only captured exactly by the time-local form of theRedfield equation. The Markovian limit would yield only asingle Lorentzian lineshape at ω = ωS1S0 + Im

∫ ∞0 dtCS1 (t),

with broadening Re∫ ∞

0 dtCS1 (t).

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