Functional Mode Singlet Fission Theory - The Journal of Physical

Subscriber access provided by NEW MEXICO STATE UNIV

Article

Functional Mode Singlet Fission Theory Justin E. Elenewski, Ulyana S. Cubeta, Edward Ko, and Hanning Chen J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 26 Jan 2017 Downloaded from http://pubs.acs.org on January 26, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Functional Mode Singlet Fission Theory Justin E. Elenewski,† Ulyana S. Cubeta,† Edward Ko‡ and Hanning Chen*† †

‡

: Department of Chemistry, the George Washington University, Washington, DC 20052, the United States of America

: Department of Mechanical Engineering, Columbia University in the City of New York, New York, NY 10027, the United States of America *: corresponding author 800 22nd Street, NW, Washington, DC 20052 Fax: 202-994-5873 Phone: 202-992-4492 Email: [email protected]

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Singlet fission is a multiple exciton generation process that splits a singlet exciton (S0S1) into a correlated triplet pair (T1T1), affording a route to overcome the longstanding Shockley-Queisser thermodynamic limit for solar energy conversion. A new theory, based on multi-configuration constrained density functional theory and functional mode analysis, has been developed to model intermolecular singlet fission in organic photovoltaics. Specifically, constrained density functional theory is first employed to construct molecular orbitals for the six spin configurations comprising T1T1, the diabatic product state. In a subsequent step, linear response time-dependent density functional theory is utilized to formulate the S0S1 diabatic reactant state. Functional mode analysis is then applied to a thermalized ensemble of diabatic energy gaps to ascertain the reaction coordinate for the S0S1→ T1T1 transition. If singlet fission is assumed to follow a direct route, its rate may be evaluated using a modified Jortner formula within strong vibronic coupling regime. In contrast, second-order perturbation theory must be adopted to treat alternate pathways that are mediated by a charge transfer (CT) intermediate. As shown through numerical simulations of single crystal tetracene, our theory reveals the direct mechanism to be the primary transition path, with an experimentally consistent singlet fission rate of 0.02 ps-1. CT pathways are effectively blocked due to a substantially diminished vibrational resonance among participating states. Our results have broad applicability, as only trivial alterations are needed to enable our new theory to model vibrationally modulated singlet fission using time-delayed pulse sequences.

2


Page 2 of 35

Page 3 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1. INTRODUCTION Singlet fission1 is a photophysical process that converts a singlet exciton into a correlated triplet pair upon the absorption of an incident photon. This phenomenon was first discovered in 1965 by S. Singh et al. when examining the multi-component decay of fluorescence intensity in anthracene crystals.2 Nevertheless, the topic was virtually neglected until a 2006 theoretical study3 invoked a new wave of interest by proposing 69 organic molecules as potential candidates for efficient singlet fission, using adiabatic excitation energies and band gap maxima as search parameters. Since singlet fission can raise the maximum incident-photon-to-converted electron (IPCE) ratio from 100% to 200%, by doubling the number of excitons, it holds great promise to circumvent the longstanding Shockley-Queisser limit4 for solar energy conversion. As an example, an internal quantum efficiency of 160% has been achieved in a 15 nm thick pentacene/fullerene heterojunction film when harvesting incident photons at 670 nm.5 In a more recent study,6 a yet higher efficiency of 170% was observed in the solutionprocessable 6,13-bis(triisopropylsilylethynyl)pentacene (TIPS-pentacene) system, which may be selectively incorporated into bandgap-modulated PbSe nanocrystals in order to facilitate productive exciton collection. Another promising strategy to promote singlet fission is to conjugate strong-acceptor and strong-donor building blocks in a polymer chain in order to stabilize the intermediate charge-transfer states that mediate the excition transition through enhanced electronic coupling. Interestingly, a femtosecond opticalpulse shaping experiment7 demonstrated that triplet state formation can be enhanced 20% by simultaneously promoting the electronic excitation and lattice motion in a polycrystalline tetracene film, suggesting a strong interplay between exciton transition

3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

and molecular vibration. This system is an ideal model for photovoltaic materials due to the presence of highly mobile, long-lived triplet excitons and a favorable energetic alignment between S0S1 and T1T1 that minimizes energy loss during photoconversion.8-9 As shown in Fig. 1, singlet fission is initiated when a chromophore is photoexcited from its ground state (S0) to the lowest-lying excited singlet configuration (S1). The excited state then donates a fraction of its energy to a nearby ground-state chromophore (S0S1), generating a correlated pair of triplet excitations (T1T1). In a final step, the spin configurations constituting the multiexciton (T1T1) state rapidly lose their correlation due to condensed phase decoherence, generating a pair of independent triplet excitons (T1+T1).10 While conceptually straightforward, the mechanism of spin conversion is not universal; the route in a given material determined by both the electronic coupling between constituent states and the overall architecture of the potential energy surface11-12. Nonetheless, the proposed pathways can be grouped into general categories arising from either direct reactant-product coupling13 or mediation by a participating charge transfer (CT) state14,15 (Fig. 2). The direct pathway is the most straightforward, affording T1T1 from S0S1 by means of the intrinsic diabatic coupling T1T1 Hˆ 0 S0 S1 that appears at first-order in perturbation theory. This mechanism becomes increasingly less applicable as the strength of the coupling becomes weak. In this case second-order perturbation theory must be invoked, thereby introducing a CT intermediate ( D0+ D1− ) to mediate singlet fission. The role of CT is defined by its energy with respect to T1T1 and S0S1. If CT lies higher in the state manifold than T1T1 or S0S1, then it will possess a vanishingly small lifetime and minimal interaction with vibrational modes. This virtual CT state simply facilitates a

4


Page 4 of 35

Page 5 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


single-step coherent resonance between S0S1 and T1T1, reminiscent of superexchange, and necessitates that all three states have large electronic couplings.16-17 Conversely, if CT and S0S1 are comparable in energy, then a direct mixing between them may promote the formation of T1T1 through strong vibronic coupling. Over the last decade, many experimental techniques have been developed and applied to probe the kinetics of singlet fission, where the timescale can range from 80 fs to 400 ps.12, 18 These data provide a fertile testbed to benchmark new theoretical methods and mechanisms. For instance, the multiexciton T1T1 may be directly detected using time-resolved two-photon photoemission spectroscopy (TR-2PPE), and its population monitored on a real-time basis by ionizing one of its constituent triplets.19 Moreover, the S0S1 and T1T1 states of the carotenoid all-trans 3R,3’R-zeaxanthin can be differentiated in distinct phases of its self-assembled aggregates using time-resolved resonance Raman spectroscopy.20 Finally, ultrafast two-dimensional electronic photon echo spectroscopy (2DES)21 can quantify the coupled electronic and vibrational dynamics of S0S1 and T1T1 excitons. This tool is particularly fascinating, as a seminal theoretical effort suggested that specific C-C stretching modes of pentacene act to promote the singlet fission process.16 Subsequent experimental studies have since established this prediction as fact, and triplet pair formation in pentacene is now known to be driven by three Raman-active vibrational modes at 265 cm-1, 1,170 cm-1 and 1360 cm-1.21 A rich library of theoretical methods have been developed to accompany these experimental endeavors, thereby permitting exploration of the delicate interplay between vertical excitation, excited state relaxation, quantum superposition and vibronic coupling in this intricate photophysical process. Among them are the multi-configuration time-

5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

dependent Hartree method (MCTDH),22 reduced density matrix dynamics (RDMD),16 trajectory surface hopping (TSH),23 symmetrical quasi-classical (SQC) approach,24 active space decomposition (ASD) strategy,25 Markovian density matrix propagation (MDMP)26, restricted active space spin-flip (RAS-SF) scheme,27 three-state kinetic model28 and constrained density functional theory-configuration interaction (CDFTCI).12, 29 While these electronic theories successfully delineate the multi-configuration characteristics of this multiple exciton generation (MEG) process, a vibrationally resolved approach is still required to directly correlate the spin-allowed transition with molecular vibrations. The need for such a method is underscored by experimental studies that extend beyond the aforementioned efforts with pentacene. In a recent “pump-andprobe” experiment,30 significant vibrational coherence transfer was observed in a TIPSpentacene thin film at five distinct phonon bands. Since the phonon frequencies extend up to 1,576 cm-1, it is technically possible to engineer the formation of correlated triplet pairs through photoexcitation with time-delayed sub-10 fs control laser pulses.31 Moreover, restricted active space calculations of potential energy profiles for tetracene derivatives32 suggest that their electronic coupling strengths are likely to undergo fluctuations commensurate with molecular vibrations.33 Thus, it is crucial to accurately determine which vibrational modes promote singlet fission when calculating its rate using adiabatic34-35 or non-adiabatic36-37 expressions under the general framework defined by Fermi’s golden rule and first-order perturbation theory.38 Although Fermi’s golden rule can be expressed in time domain, its energy-dependent version is more computationally convenient as only a handful of energetic and kinetic parameters are needed. This makes it particularly suitable for modeling singlet fission, where the timescale can vary over

6


Page 6 of 35

Page 7 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


several orders of magnitude.12 For example, singlet fission in tetracene single crystals occurs at a rather sluggish rate of 11 ns-1,39 which, to the best of our knowledge, is computationally inaccessible by all current time-dependent ab initio approaches. Inspired by functional mode analysis,40 we formulated a functional mode electron transfer theory (FMET)41 that is applicable to arbitrary vibronic coupling strengths. More recently, the FMET was successfully extended to treat hot electron transfer under nonthermalized conditions.42 Specifically, the electron injection rate from a photo-excited 6methyl-azulene-2-carboxylic acid dye molecule into anatase TiO2 [101] surface was found to be a function of the incident photon wavelength.42 In the present study, our FMET will be further developed to model singlet fission, entailing multiple spin configurations that participate in a MEG process. The remainder of this paper is organized as follows. In section 2, the FMET is briefly reviewed before introducing constrained density functional theory (CDFT)43 to construct molecular orbitals for the six constituent spin configurations of the correlated triplet pair T1T1. The reaction coordinate for the S0S1→T1T1 transition is then ascertained through functional mode analysis by projecting the diabatic energy gap onto vibrational normal modes. Using thermodynamic integration, the potential energy surface of the spin-allowed transition is likewise constructed in order to calculate the free energy driving force and reorganization energy. The electronic coupling strength between S0S1 and T1T1, is likewise quantified by using an orbital reorthogonalization method44 that maximally localizes the diabatic states. Finally, the singlet fission rate is evaluated using a revised Jortner formula41 that includes contributions from all vibrational modes that lead to formation of the correlated triplet pair. In section 3, our proposed functional mode singlet fission (FMSF) theory is justified

7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 35

through numerical simulations of a tetracene single crystal, a well-studied model system. When taken together with experimental data,19, 39 these calculations reveal a clear path toward singlet fission in tetracene. Finally, we will summarize our results and succinctly discuss the possible applications of our new theory in section 4.

2. THEORETICAL METHODS !

As detailed in Ref. 41, the reaction coordinate, VMCM , of a given electron-transfer or spin-transition process between donor, Ψ D , and acceptor, Ψ A , diabatic states can be formulated as a linear combination of all vibrational normal modes: N vib ! ! VMCM = ∑ ciVi

(1)

i=1

where the expansion coefficient, ci , indicates the relative contribution of the i th !

vibrational mode, Vi , to the degeneracy of vibronic energy that is required for a nonradiative transition. In the present study, the vibrational modes were determined by diagonalizing the mass-weighted cross-correlation matrix of atomic displacements in a molecular dynamics trajectory. By following the functional mode analysis method,40 ! VMCM is ascertained through maximization of the Pearson’s correlation coefficient, R ,

which is defined as R=

where pV!

MCM

!

(t ) = ( r (t ) −

(

cov pV!MCM (t), ΔE DA ( t )

σ V!MCM σ ΔEDA

)

(2)

! ! r ) i VMCM is the projection of instantaneous nuclear configuration

! ! r ( t ) on VMCM , ΔE DA ( t ) = E D ( t ) − E A ( t ) is the instantaneous diabatic energy gap,

8


Page 9 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(

)

cov pV!MCM (t), ΔE DA ( t ) is the covariance function between pV!MCM and ΔE DA , with their

standard deviations denoted as σ V!

and σ ΔE , respectively. The search for the DA

MCM

maximum value of R reduces to solving a set of linearly coupled equation:

∑ c cov( p N vib

i

i=1

! Vi (t )

)

(

, pV!j (t ) = cov ΔE DA ( t ) , pV!j (t )

)

j = 1,..., N vib

(3)

Due to the multiexciton character of the acceptor T1T1 (Fig. 1), it is constructed through a multiconfigurational CDFT-based approach.43,

45

To be specific, a spin-

polarized Hartree potential was applied to a pair of adjacent chromophore molecules to achieve a net spin difference of +4 (or -4) between them as quantified by Becke population analysis.46 The resulting molecular orbitals were then adopted as building blocks to construct the six constituent spin-adapted determinants of T1T1 state through spin flips. Construction of the donor S0S1 state was initiated by using linear-response time-dependent density functional theory (LR-TDDFT) to diagonalize a response matrix involving all occupied-virtual orbital pairs.47 Then, the collective excitations of those orbital pairs were simplified to a compact particle-hole transition using the natural transition orbitals (NTO) transformation.48 Finally, the resultant highest occupied NTO (HONTO) and the lowest unoccupied NTO (LUNTO) were adopted to form the S0S1 state as a linear combination of two spin-adapted determinants. Because of thermodynamic reversibility (Fig. 3), the free energy gap between the S0S1 and T1T1 states, ΔGS S →T T is given by 0 1

1 1

(

ΔGS0S1→T1T1 = ΔGS0S0 →T1T1 − ΔES0S0 →S0S 1 − ΔETR

)

(4)

where ΔGS S →T T is the gap between S0S0 and T1T1, ΔES S →S S is the vertical excitation 0 0

1 1

0 0

0 1

energy and ΔETR is the thermal relaxation energy at S0S1. Although it is trivial to 9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 35

calculate both ΔES S →S S and ΔETR using LR-TDDFT,47 the thermodynamic integration 0 0

0 1

method49 must be utilized to compute ΔGS S →T T through a linearly mixed Hamiltonian, 0 0

1 1

H mix (η ) , that spans over eleven sampling windows of 10-ps each with evenly spaced

values of η : H mix (η ) = (1− η ) Hˆ S0S0 + η Hˆ T1T1

(5)

After sampling the gradient of H mix (η ) along η , ΔGS S →T T is obtained by discrete 0 0

1 1

summation: ⎛ ∂H mix (η ) ⎞ ΔGS0S0 →T1T1 = ∑ ⎜ Δηi ∂η ⎟⎠ ηi i=1 ⎝ Nη

(6)

Without any additional simulation, the reorganization energy, λ , can be also deduced: λ = H mix (η = 1) − H mix (η = 0 )

η =0

+ ΔGS0S0 →T1T1

(7)

Moreover, using the extended Slater-Condon rule50 for non-orthonormal molecular orbitals (see Supporting Information), T1T1 Hˆ 0 S0 S1 is simplified to the sum of couplings between selected spin configurations: ⎛ 3 ⎜ T1T1 Hˆ 0 S0 S1 = 2⎜ ⎜⎝

↓ ↓

↑ Hˆ 0 ↑

↑ ↑↓

+

↓

↑ ↑

↓ Hˆ 0 ↓

↑ ↑↓

↓

⎞ ⎟ ⎟ ⎟⎠

(8)

In order to ensure the rigorous orthogonality between T1T1 and S0S1, a revised expression44 that affords orbital renormalization is adopted:

10


Page 11 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


T1T1 Hˆ 0 S0 S1 = ⎛ ⎜ ETS = ⎜ ⎜⎝

where

⎛ ⎜ STS = ⎜ ⎜⎝

↓ ↓

and ET =

↑

↑ ↑

↑↓

↓ ↓

↑ Hˆ 0 ↑

+

↓

↓ ↓

↑ ↑

(

↑ Hˆ 0 ↑

↓ ↓

↑ ↑

3 ⎛ 2ETS − STS ( ET + ES ) ⎞ ⎜ ⎟ 2 2⎝ 2 1− STS ⎠

↓ ↓

or

↑ ↑↓

↑ ↑↓

↑ ↑

↓

)

+

↑ ↑

↓

⎞ ⎟, E = S ⎟ ⎟⎠

↓ Hˆ 0 ↓

↑ ↑

↓ ↓

(9)

↑

↓ Hˆ 0 ↓

↑↓

↑ ↑↓

↓

↓

⎞ ⎟ ⎟ ⎟⎠

,

↑

Hˆ 0

, ↑↓

↓

.

Once the free energy gap, ΔG0 , reorganization energy, λ , and electronic coupling strength, J , associated with the S0S1→T1T1 transition are all determined, we are ready to evaluate the overall singlet fission rate, kSF , using either the Jortner formula for strong vibronic coupling ( λ ≫ "ω MCM ) or the energy-gap law for weak vibronic coupling ( λ ≪ "ω MCM ).41

3. SIMULATION RESULTS The singlet fission rates of tetracene crystals and thin films are known to be uniformly slow, with values ranging from 90 ps to 300 ps in crystalline samples and increasing by roughly an order of magnitude to 22 ps for polycrystalline thin-films39, 51-52. This corresponds to rate constants kSF that range between ~0.003 ps-1 and ~0.011 ps-1 in the single crystal case, with thin-film data indicating stability over temperatures spanning from 10 K to 270 K. Nevertheless, a thermal activation energy of 0.18 eV was estimated from a prompt fluorescence emission experiment,53 contradicting the temperature 11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 35

independence of kSF . In a recent TR-2PPE study,19 the slight endothermicity is presumed to be surmountable by entropic gain due to the increased number of exciton molecules and spin configurations during the formation of correlated triplet pairs. Moreover, a miniscule activation energy of 0.06 eV was deduced from the temperature dependence of the prompt fluorescence decay rate,54 making the S0S1 and T1T1 states nearly thermodynamically degenerate at room temperature. Tetracene single crystals are thus an ideal system to unravel the enduring mystery of sluggish singlet fission in the presence of a diminished energy barrier. In our study, an 8 × 10 × 5 tetracene supercell (Fig. 4) with a total of 24,000 atoms was constructed from an experimental X-ray crystal structure.55 This supercell was then equilibrated at 300 K using a 1-ns molecular dynamics (MD) simulation with a timestep of 1.0 fs and the GAFF force field56 and RESP partial charge fitting scheme.57 Thereafter, a 5-ns MD production run was carried out to provide 50,000 randomly selected snapshots for normal mode analysis40 on an adjacent pair of tetracene molecules (Fig. 4) chosen for their shortest center-of-mass distance. This designated pair also serves as our simulated reaction center for singlet fission. In order to achieve a balance between physical accuracy and numerical efficiency, the hybrid quantum mechanics/molecular mechanics (QM/MM) approach58 was adopted to model the singlet fission within the reaction center using density functional theory, while the thermal fluctuations of other molecules were described by empirical force fields. Unless otherwise specified, all QM/MM simulations were performed using the CP2K software59 with the Goedecker-Teter-Hutter (GTH) pseudopotential,60

Perdew-Burke-Ernzehof

12

hybrid

(PBE0)


exchange-correlation

Page 13 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


functional,61 polarized-valence-double- ζ (PVDZ) basis set,62 electrostatic QM/MM coupling scheme,63 and a wavelet-based Poisson solver.64 The success of our new theory rests on an accurate determination of the collective !

vibrational mode, VMCM , that drives singlet fission between the S0S1 and T1T1 states. To this end, 5,000 nuclear configurations were randomly extracted from our 5-ns MD production trajectory to form a well-equilibrated ensemble for FMA; a procedure that projects their diabatic energy gaps onto all vibrational normal modes of the chosen molecular pair. As shown in Fig. 5, a concerted ring-deformation motion was found to be !

the VMCM with an effective wavenumber ω MCM of 1588 cm-1, calculated as N vib

−1 ω MCM = ∑ ci2ω i−1 . Of the 174 available vibrational modes, only 4 are essential to singlet i=1

!"

fission as indicated by the profile of V MCM coefficients (Fig. 5) with ci2 > 0.05 . In contrast, the remaining 170 vibrational modes only contribute a cumulative statistical !

!

weight of 28.6% to VMCM . A further decomposition of VMCM reveals the four essential modes, which are individually illustrated in Fig. 6. The greatest contribution, comprising !"

approximately 30% of V MCM , comes from a 1788 cm-1 ring-stretching mode (Fig. 6a) that rests predominantly on only one of the two tetracene molecules. The second and third !

most significant modes have comparable contributions of ~15% to VMCM , and consist of a 1353 cm-1 C-H shearing mode (Fig. 6b) and a 1685 cm-1 ring-breathing mode (Fig. 6c), respectively. An additional ring-shearing mode (Fig. 6d) at 1788 cm-1 only contributes 9%, yet interestingly exhibits a similar type of motion. Previous investigations have indicated that slow vibrational modes may collude to enhance the singlet fission process, presumably by bringing adjacent tetracene units closer together and thus enhancing their

13



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 35

diabatic coupling.7 The high-frequency modes identified through our functional mode analysis further promote singlet fission by means of the Franck-Condon effect, maximizing the overlap between vibrational wavefunctions. These four essential vibrational modes are substantially faster than thermal fluctuations at ambient conditions, i.e., !ω ≫ kBT , suggesting a weak temperature dependence of the singlet fission rate. As !

a prudent measure to ensure the statistical quality of VMCM , the coefficients ci2 were projected onto another 3,000 nuclear configurations that were extracted from a separate 3-ns MD trajectory. As shown in Fig. 7, the cross-validated Pearson’s coefficient RC is 0.80, which is very close to the training coefficient R of 0.85. The small deviation of RC from R signifies sufficient sampling for FMA. The energy diagram shown in Fig. 3 was calculated by thermodynamic integration49 and LR-TDDFT.47 LR-TDDFT was first employed to compute the S0S0→S0S1 vertical excitation energy for 5,000 MD snapshots, revelaing a peak centered at 2.57 eV with a standard deviation of 0.08 eV (Fig. 8). Next, ΔGS S →T T was evaluated 0 0

1 1

by thermodynamic integration to obtain an energy of 2.02 eV. After accounting for a thermal relaxation energy of ΔEER = 0.18 eV in the S0S1 state, the driving force associated with singlet fission, −ΔGS S →T T = ΔES S →S S − ΔEER − ΔGS S →T T , was found to be 0.38 eV, 0 1

1 1

0 0

0 1

0 0

1 1

which is only slightly smaller than the reorganization energy λ = 0.57 eV. Surprisingly, the S0S1 and T1T1 states are nearly energetically degenerate upon vertical excitation, eliminating the need for thermal activation as suggested by a transient absorption spectroscopy study.39 However, due to the comparatively small electronic coupling strength J = T1T1 Hˆ 0 S0 S1 (as will be discussed later), singlet fission cannot proceed

14


Page 15 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


through a mechanism in which the constituent states form a coherent superposition. In an alternate pathway, the vertically excited S0S1 state relaxes to its optimal geometry before sneaking across the T1T1 potential energy surface through vibrational quantum tunneling, in the presence of strong vibronic coupling ( λ ≫ "ω MCM ) and a fast driving mode ( 41 !ω MCM ≫ k BT ). In this case, the revised Jorner formula must be chosen to calculate the

singlet fission rate, kSF : 1 ⎛ ΔG0 + λ ⎞ !ω F ⎟⎠

2π J 2 − 2 ⎜⎝ kSF = 2 e ! ωF

2

(10)

where ω F is the functional-weighted vibrational angular frequency defined as ω F2 =

λ N vib 2 ∑ ci ω i ! i=1

(11)

The only undetermined parameter in Eq. 10 is now the electronic coupling strength J , which is given by Eq. 9. Since J fluctuates with structural changes, its mean value of 1.47 meV was determined by sampling 100 MD snapshots. To access basis-set completeness, J was also calculated with the more diffuse aug-cc-pVDZ basis set62 and an adjacent molecular pair in the crystal structure, affording a rather consistent value of 1.17 meV. In fact, values of J between 1.0 and 2.0 meV were obtained in another multistate DFT simulation15 when using an array of 56 tetracene monomers and the same PBE0 functional employed in our calculations.61 With all parameters (ΔG0 = −0.38 eV , λ = 0.57 eV , ω F = 2763 cm −1 and J = 1.47 meV) in hand, Eq. 10 was utilized to calculate

the rate of singlet fission, kSF , assuming that it follows the direct mechanism (Fig. 2). Our calculated kSF is found to be 0.020 ps-1, in qualitative agreement with the experimental rate of 0.011 ps-1.39 Despite this slight overestimation, a direct S0S1 → T1T1 transition is

15



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 35

found to be completely feasible as a result of strong vibrational tunneling, 1 !ω F ≈ ΔG0 + λ . That is, the vibrational zero-point energy is sufficient to provide the 2

energy needed by the reactant S0S1 state to reach its crossing point with the T1T1 product state. To explore the feasibility of the CT-mediated pathway (Fig. 2), the relative energetics of CT state were calculated with respect to S0S1 and T1T1, alongside the electronic coupling strengths J CT −S S and J CT −T T . Since the dipolar CT state is known to 0 1

1 1

be stabilized by electronic polarization in its surrounding environment, the shift in its energy with respect to the ground state was explored by including all eight molecules in the first neighbor shell of a dimer into QM subsystem (Fig. 4). As shown in Table 1, ΔES0S0 →CT is dramatically reduced from 4.01 eV to 2.74 eV, while the later value is

consistent with previous experimental and theoretical findings.65-67 The same downward shift in the CT state’s energy upon expansion of the QM subsystem has been observed in two other recent DFT studies.29, 68 In contrast, ΔES S →T T is nearly invariant at ~2.0 eV 0 0

1 1

due to the weak electric dipole moment of the T1T1 state. This suggests that a sequential S0S1 → CT → T1T1 route is effectively blocked since ES S < ECT 0 1

.

In contrast, the

superexchange-like mediated mechanism is defined by a manifold of virtual CT vibronic states, and its relative contribution to kSF versus the direct mechanism can be estimated using second-order perturbation theory, which explicitly accounts for the mixing of CT and S1S0 states through vibronic coupling:69 kcoherent ⎛ J CT −S0S1 J CT −T1T1 ⎞ =⎜ kdirect J S0S1 −T1T1 ⎟⎠ ⎝

2

∑∑ S0 S1 CT

2

vS0S1 vCT vS0S1 vT1T1

16

2

(

vCT vT1T1

2

ECT − ES0S1


)

2

(

δ ES0S1 − ET1T1

)

(12)

Page 17 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


where the v ’s correspond to vibrational eigenstates of the corresponding functional modes. Since !ω MCM ≫ kBT , only the ground vibrational state of S0S1 is considered in our calculation, and vCT and vT T are represented by the well-known analytic solutions for 1 1

displaced, one-dimensional harmonic oscillators. Even if the minimum of the CT potential energy surface is assumed to lie directly above the crossing point between those of S0S1 and T1T1 in order to maximize the vibrational wavefunction overlap, the upper limit of the estimated

kcoherent is given by ~0.30 as a result of substantial vibrational kdirect

decoherence. Thus, a CT-mediated mechanism is unlikely to be the primary route for singlet fission in single crystal tetracene as the enhanced electronic coupling strengths

( i.e., J

CT −T1T1

)

= 31.5 meV > J CT −S0S1 = 6.30 meV > J S0S1 −T1T1 = 1.47 meV are outweighed by the

diminished vibrational resonance.

4. DISCUSSION AND CONCLUSIONS Singlet fission has received increasing attention over the past decade as a promising implementation of MEG in environmentally friendly and cost-effective dyesensitized solar cells. In an ideal singlet fission photovoltaic cell, formation of the triplet pair must outpace a series of competing processes such as charge separation, fluorescence emission and triplet-triplet annihilation. As a consequence, a key ingredient for efficacious conversion of the photo-excited S0S1 state to the correlated T1T1 pair is strong vibronic coupling in the presence of multiple spin configurations. This strong coupling can stem from a direct interaction between reactant and product states, or it may be mediated by an intermediate CT configuration. The first pathway is exemplified by

17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 35

covalent pentacene dimers13, in which a sizable mixing between S0S1 and T1T1 eliminates the need for a CT intermediate. In contrast, block copolymers with strong-donor / strongacceptor moieties are characterized70 by an interconversion that proceeds through a stabilized CT state, affording a quantum yield of 170% for singlet fission. Irrespective of the detailed mechanism, both studies13,

70

suggest that singlet fission is driven by

molecular vibrational modes that are substantially faster than thermal fluctuations. Inspired by these experimental findings, we have extended our FMET theory to treat singlet fission using multi-configuration CDFT. As shown by our numerical simulations !

of single crystal tetracene, a fast collective vibrational mode VMCM promotes singlet fission

via

the

direct

mechanism

with

an

effective

angular

frequency

of

ω MCM = 1588 cm −1 . This occurs despite the weak electronic coupling strength of 1.47

meV. Although the electronic coupling can be modestly enhanced through coherent charge transfer dynamics, a substantial loss of vibrational wavefunction overlap ultimately disables a CT-mediated route for singlet fission. Taken together, these data suggest that direct production of a coherent triplet pair is the dominant route for singlet fission in single crystal tetracene, consistent with quantum beat spectroscopy.54 These results are similar to recent observations in covalently-linked pentancene dimers,13 though other theoretical investigations imply that an admixture of mechanisms may be operative.71 Our calculated kSF is likewise in good agreement with experimental data,39 justifying the use of our theory for the systematic design of singlet-fission solar cells. We should note that our calculations are performed on an isolated dimer, while a proper treatment necessitates the use of a fully periodic model in order to accurately quantify delocalization of singlet and triplet states.17 The solid agreement between our calculations

18


Page 19 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and experiment is nonetheless remarkable, and may be put into context through subsequent investigations. In a broader context, our FMA-based method can be readily adapted to model triplet fusion in organic light-emitting diodes (OLED),72 another important class of materials for sustainable energy applications.

Supporting Information: Expectation values for the electronic coupling strength between the multiconfigurational S0S1 and T1T1 states.

19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Acknowledgements The research was supported by a start-up grant and the Columbian College Facilitating Fund of the George Washington University. Computational resources utilized in this research were provided by the Argonne Leadership Computing Facility (ALCF) at Argonne National Laboratory under Department of Energy contract DE-AC0206CH11357 and by the Extreme Science and Engineering Discovery Environment (XSEDE) at Texas Advanced Computing Center under National Science Foundation contract TG-CHE130008.

20


Page 20 of 35

Page 21 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figures:

Figure 1. Flowchart for singlet fission, illustrating all participating electronic states and the constituent spin configurations of their frontier orbitals. The red bar indicates the lowest unoccupied orbital or the higher-energy singly occupied orbital, while the blue bar represents the highest doubly occupied orbital or the lower-energy singly occupied orbital.

21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 2. Proposed mechanisms for singlet fission, namely the direct (solid line) and charge-transfer state mediated (dotted line) mechanisms. In the constituent spin configurations of the intermediate charge-transfer (CT) state, D0+ D1− , the red bar indicates the lowest unoccupied orbital or the higher-energy singly occupied orbital while the blue bar represents the highest doubly occupied orbital or the lower-energy singly occupied orbital.

22


Page 22 of 35

Page 23 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3. The energy diagram for singlet fission in single crystal tetracene.

23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 4. An 8 × 10 × 5 supercell of a tetracene single crystal. The molecular pair designated as the reaction center for singlet fission is highlighted in red, while all eight molecules consisting of its first neighbor shell in the ab crystalline plane are highlighted in blue.

24


Page 24 of 35

Page 25 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


!

Figure 5. Calculated driving vibrational mode, VMCM , for singlet fission within an adjacent pair of tetracene molecules.

25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 6. The four vibrational normal modes essential to singlet fission alongside their wavenumbers and MCM coefficients.

26


Page 26 of 35

Page 27 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 7. Scatter plot for cross-validation of MCM coefficients, ci2 , by projecting the diabatic energy gap, ΔES S →T T onto all vibrational normal modes. 0 1

1 1

27



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 8. Histogram of vertical excitation energies between the S0S0 and S0S1 states.

28


Page 28 of 35

Page 29 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Tables: Cluster

Dimer

Decamer

ΔES0S0 →T1T1 (eV)

2.04

2.01

ΔES0S0 →CT (eV)

4.01

2.74

Table 1. The shift in energy levels for the T1T1 and CT states with respect to the ground S0 S0 state upon expanding the QM subsystem by including the first neighbor shell of the designated dimer as shown in Fig. 4.

29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

References: 1. Smith, M. B.; Michl, J., Singlet Fission. Chem. Rev. 2010, 110, 6891-6936. 2. Singh, S.; Jones, W. J.; Siebrand, W.; Stoicheff, B. P.; Schneider, W. G., Laser Generation of Excitons and Fluorescence in Anthracene Crystals. J. Chem. Phys. 1965, 42, 330-342. 3. Paci, I.; Johnson, J. C.; Chen, X.; Rana, G.; Popović, D.; David, D. E.; Nozik, A. J.; Ratner, M. A.; Michl, J., Singlet Fission for Dye-Sensitized Solar Cells: Can a Suitable Sensitizer Be Found? J. Am. Chem. Soc. 2006, 128, 16546-16553. 4. Shockley, W.; Queisser, H. J., Detailed Balance Limit of Efficiency of P-N Junction Solar Cells. J. Appl. Phys. 1961, 32, 510-519. 5. Congreve, D. N.; Lee, J.; Thompson, N. J.; Hontz, E.; Yost, S. R.; Reusswig, P. D.; Bahlke, M. E.; Reineke, S.; Van Voorhis, T.; Baldo, M. A., External Quantum Efficiency above 100% in a Singlet-Exciton-Fission–Based Organic Photovoltaic Cell. Science 2013, 340, 334-337. 6. Yang, L.; Tabachnyk, M.; Bayliss, S. L.; Böhm, M. L.; Broch, K.; Greenham, N. C.; Friend, R. H.; Ehrler, B., Solution-Processable Singlet Fission Photovoltaic Devices. Nano Lett. 2015, 15, 354-358. 7. Grumstrup, E. M.; Johnson, J. C.; Damrauer, N. H., Enhanced Triplet Formation in Polycrystalline Tetracene Films by Femtosecond Optical-Pulse Shaping. Phys. Rev. Lett. 2010, 105, 257403-257406. 8. Akselrod, G. M.; Deotare, P. B.; Thompson, N. J.; Lee, J.; Tisdale, W. A.; Baldo, M. A.; Menon, V. M.; Bulović, V., Visualization of Exciton Transport in Ordered and Disordered Molecular Solids. Nat. Commun. 2014, 5, 3646. 9. Wan, Y.; Guo, Z.; Zhu, T.; Yan, S.; Johnson, J.; Huang, L., Cooperative Singlet and Triplet Exciton Transport in Tetracene Crystals Visualized by Ultrafast Microscopy. Nat Chem 2015, 7, 785-792. 10. Scholes, G. D., Correlated Pair States Formed by Singlet Fission and Exciton– Exciton Annihilation. J. Phys. Chem. A. 2015, 119, 12699-12705. 11. Smith, M. B.; Michl, J., Recent Advances in Singlet Fission. Annu. Rev. Phys. Chem. 2013, 64, 361-386. 12. Yost, S. R., et al., A Transferable Model for Singlet-Fission Kinetics. Nat Chem 2014, 6, 492-497. 13. Fuemmeler, E. G., et al., A Direct Mechanism of Ultrafast Intramolecular Singlet Fission in Pentacene Dimers. ACS Cent. Sci. 2016, 2, 316-324. 14. Monahan, N.; Zhu, X. Y., Charge Transfer–Mediated Singlet Fission. Annu. Rev. Phys. Chem. 2015, 66, 601-618. 15. Chan, W.-L.; Berkelbach, T. C.; Provorse, M. R.; Monahan, N. R.; Tritsch, J. R.; Hybertsen, M. S.; Reichman, D. R.; Gao, J.; Zhu, X. Y., The Quantum Coherent Mechanism for Singlet Fission: Experiment and Theory. Acc. Chem. Res. 2013, 46, 13211329. 16. Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R., Microscopic Theory of Singlet Exciton Fission. I. General Formulation. J. Chem. Phys. 2013, 138, 114102114117.

30


Page 30 of 35

Page 31 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


17. Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R., Microscopic Theory of Singlet Exciton Fission. III. Crystalline Pentacene. J. Chem. Phys. 2014, 141, 074705074716. 18. Chan, W.-L.; Ligges, M.; Jailaubekov, A.; Kaake, L.; Miaja-Avila, L.; Zhu, X.Y., Observing the Multiexciton State in Singlet Fission and Ensuing Ultrafast Multielectron Transfer. Science 2011, 334, 1541-1545. 19. Chan, W.-L.; Ligges, M.; Zhu, X. Y., The Energy Barrier in Singlet Fission Can Be Overcome through Coherent Coupling and Entropic Gain. Nat Chem 2012, 4, 840845. 20. Wang, C.; Tauber, M. J., High-Yield Singlet Fission in a Zeaxanthin Aggregate Observed by Picosecond Resonance Raman Spectroscopy. J. Am. Chem. Soc. 2010, 132, 13988-13991. 21. Bakulin, A. A.; Morgan, S. E.; Kehoe, T. B.; Wilson, M. W. B.; Chin, A. W.; Zigmantas, D.; Egorova, D.; Rao, A., Real-Time Observation of Multiexcitonic States in Ultrafast Singlet Fission Using Coherent 2D Electronic Spectroscopy. Nat Chem 2016, 8, 16-23. 22. Tamura, H.; Huix-Rotllant, M.; Burghardt, I.; Olivier, Y.; Beljonne, D., FirstPrinciples Quantum Dynamics of Singlet Fission: Coherent Versus Thermally Activated Mechanisms Governed by Molecular 𝜋-Stacking. Phys. Rev. Lett. 2015, 115, 107401. 23. Akimov, A. V.; Prezhdo, O. V., Nonadiabatic Dynamics of Charge Transfer and Singlet Fission at the Pentacene/C60 Interface. J. Am. Chem. Soc. 2014, 136, 1599-1608. 24. Tao, G., Understanding Electronically Non-Adiabatic Relaxation Dynamics in Singlet Fission. J. Chem. Theory Comput. 2015, 11, 28-36. 25. Parker, S. M.; Seideman, T.; Ratner, M. A.; Shiozaki, T., Model Hamiltonian Analysis of Singlet Fission from First Principles. J. Phys. Chem. C. 2014, 118, 1270012705. 26. Mirjani, F.; Renaud, N.; Gorczak, N.; Grozema, F. C., Theoretical Investigation of Singlet Fission in Molecular Dimers: The Role of Charge Transfer States and Quantum Interference. J. Phys. Chem. C. 2014, 118, 14192-14199. 27. Zimmerman, P. M.; Musgrave, C. B.; Head-Gordon, M., A Correlated Electron View of Singlet Fission. Acc. Chem. Res. 2013, 46, 1339-1347. 28. Kolomeisky, A. B.; Feng, X.; Krylov, A. I., A Simple Kinetic Model for Singlet Fission: A Role of Electronic and Entropic Contributions to Macroscopic Rates. J. Phys. Chem. C. 2014, 118, 5188-5195. 29. Turban, D. H. P.; Teobaldi, G.; O'Regan, D. D.; Hine, N. D. M., Supercell Convergence of Charge-Transfer Energies in Pentacene Molecular Crystals from Constrained Dft. Phys. Rev. B 2016, 93, 165102. 30. Musser, A. J.; Liebel, M.; Schnedermann, C.; Wende, T.; Kehoe, T. B.; Rao, A.; Kukura, P., Evidence for Conical Intersection Dynamics Mediating Ultrafast Singlet Exciton Fission. Nat Phys 2015, 11, 352-357. 31. Fischer, A., et al., Molecular Wave-Packet Dynamics on Laser-Controlled Transition States. Phys. Rev. A. 2016, 93, 012507. 32. Casanova, D., Electronic Structure Study of Singlet Fission in Tetracene Derivatives. J. Chem. Theory Comput. 2014, 10, 324-334.

31



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

33. Alguire, E. C.; Subotnik, J. E.; Damrauer, N. H., Exploring Non-Condon Effects in a Covalent Tetracene Dimer: How Important Are Vibrations in Determining the Electronic Coupling for Singlet Fission? J. Phys. Chem. A. 2015, 119, 299-311. 34. Holstein, T., Studies of Polaron Motion: Part II. The “Small” Polaron. Ann. Phys. 1959, 8, 343-389. 35. Holstein, T., Studies of Polaron Motion: Part I. The Molecular-Crystal Model. Ann. Phys. 1959, 8, 325-342. 36. Marcus, R. A., Chemical and Electrochemical Electron-Transfer Theory. Annu. Rev. Phys. Chem. 1964, 15, 155-196. 37. Englman, R.; Jortner, J., The Energy Gap Law for Radiationless Transitions in Large Molecules. Mol. Phys. 1970, 18, 145-164. 38. Dirac, P. A. M., The Quantum Theory of the Emission and Absorption of Radiation. Proc. Roy. Soc. A. 1927, 114, 243-265. 39. Wilson, M. W. B.; Rao, A.; Johnson, K.; Gélinas, S.; di Pietro, R.; Clark, J.; Friend, R. H., Temperature-Independent Singlet Exciton Fission in Tetracene. J. Am. Chem. Soc. 2013, 135, 16680-16688. 40. Hub, J. S.; de Groot, B. L., Detection of Functional Modes in Protein Dynamics. PLoS Comput Biol 2009, 5, e1000480. 41. Chen, H., Functional Mode Electron-Transfer Theory. J. Phys. Chem. B. 2014, 118, 7586-7593. 42. Elenewski, J. E.; Cai, J. Y.; Jiang, W.; Chen, H., Functional Mode Hot Electron Transfer Theory. J. Phys. Chem. C. 2016, 120, 20579-20587. 43. Wu, Q.; Van Voorhis, T., Direct Optimization Method to Study Constrained Systems within Density-Functional Theory. Phys. Rev. A. 2005, 72, 024502-024505. 44. Farazdel, A.; Dupuis, M.; Clementi, E.; Aviram, A., Electric-Field Induced Intramolecular Electron Transfer in Spiro .Pi.-Electron Systems and Their Suitability as Molecular Electronic Devices. A Theoretical Study. J. Am. Chem. Soc. 1990, 112, 42064214. 45. Wu, Q.; Kaduk, B.; Van Voorhis, T., Constrained Density Functional Theory Based Configuration Interaction Improves the Prediction of Reaction Barrier Heights. J. Chem. Phys. 2009, 130, 034109-034115. 46. Becke, A. D., A Multicenter Numerical Integration Scheme for Polyatomic Molecules. J. Chem. Phys. 1988, 88, 2547-2553. 47. Casida, M. E.; Huix-Rotllant, M., Progress in Time-Dependent DensityFunctional Theory. Annu. Rev. Phys. Chem. 2012, 63, 287-323. 48. Martin, R. L., Natural Transition Orbitals. J. Chem. Phys. 2003, 118, 4775-4777. 49. Kirkwood, J. G., Statistical Mechanics of Fluid Mixtures. J. Chem. Phys. 1935, 3, 300-313. 50. Verbeek, J.; Van Lenthe, J. H., Advances in Valence Bond Theoryon the Evaluation of Non-Orthogonal Matrix Elements. J. Mol. Struc. THEOCHEM 1991, 229, 115-137. 51. Arias, D. H.; Ryerson, J. L.; Cook, J. D.; Damrauer, N. H.; Johnson, J. C., Polymorphism Influences Singlet Fission Rates in Tetracene Thin Films. Chem. Sci. 2016, 7, 1185-1191. 52. Piland, G. B.; Bardeen, C. J., How Morphology Affects Singlet Fission in Crystalline Tetracene. J. Phys. Chem. Lett. 2015, 6, 1841-1846.

32


Page 32 of 35

Page 33 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


53. Tomkiewicz, Y.; Groff, R. P.; Avakian, P., Spectroscopic Approach to Energetics of Exciton Fission and Fusion in Tetracene Crystals. J. Chem. Phys. 1971, 54, 45044507. 54. Burdett, J. J.; Bardeen, C. J., Quantum Beats in Crystalline Tetracene Delayed Fluorescence Due to Triplet Pair Coherences Produced by Direct Singlet Fission. J. Am. Chem. Soc. 2012, 134, 8597-8607. 55. Robertson, J. M.; Sinclair, V. C.; Trotter, J., The Crystal and Molecular Structure of Tetracene. Acta Cryst. 1961, 14, 697-704. 56. Wang, J.; Wolf, R. M.; Caldwell, J. W.; Kollman, P. A.; Case, D. A., Development and Testing of a General Amber Force Field. J. Comput. Chem. 2004, 25, 1157-1174. 57. Bayly, C. I.; Cieplak, P.; Cornell, W.; Kollman, P. A., A Well-Behaved Electrostatic Potential Based Method Using Charge Restraints for Deriving Atomic Charges: The Resp Model. J. Phys. Chem. 1993, 97, 10269-10280. 58. Warshel, A.; Levitt, M., Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and Steric Stabilization of the Carbonium Ion in the Reaction of Lysozyme. J. Mol. Biol. 1976, 103, 227-249. 59. VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J., Quickstep: Fast and Accurate Density Functional Calculations Using a Mixed Gaussian and Plane Waves Approach. Comput. Phys. Commun. 2005, 167, 103-128. 60. Goedecker, S.; Teter, M.; Hutter, J., Separable Dual-Space Gaussian Pseudopotentials. Phys. Rev. B 1996, 54, 1703-1710. 61. Perdew, J. P.; Ernzerhof, M.; Burke, K., Rationale for Mixing Exact Exchange with Density Functional Approximations. J. Chem. Phys. 1996, 105, 9982-9985. 62. Woon, D. E.; Dunning, J. T. H., Gaussian Basis Sets for Use in Correlated Molecular Calculations. IV. Calculation of Static Electrical Response Properties. J. Chem. Phys. 1994, 100, 2975-2988. 63. Laino, T.; Mohamed, F.; Laio, A.; Parrinello, M., An Efficient Real Space Multigrid Qm/Mm Electrostatic Coupling. J. Chem. Theory Comput. 2005, 1, 1176-1184. 64. Genovese, L.; Deutsch, T.; Neelov, A.; Goedecker, S.; Beylkin, G., Efficient Solution of Poisson’s Equation with Free Boundary Conditions. J. Chem. Phys. 2006, 125, 074105-074109. 65. Sebastian, L.; Weiser, G.; Bässler, H., Charge Transfer Transitions in Solid Tetracene and Pentacene Studied by Electroabsorption. Chem. Phys. 1981, 61, 125-135. 66. Bounds, P. J.; Siebrand, W.; Eisenstein, I.; Munn, R. W.; Petelenz, P., Calculation and Spectroscopic Assignment of Charge-Transfer States in Solid Anthracene, Tetracene and Pentacene. Chemical Physics 1985, 95, 197-212. 67. Yamagata, H.; Norton, J.; Hontz, E.; Olivier, Y.; Beljonne, D.; Brédas, J. L.; Silbey, R. J.; Spano, F. C., The Nature of Singlet Excitons in Oligoacene Molecular Crystals. The Journal of Chemical Physics 2011, 134, 204703-204713. 68. Aragó, J.; Troisi, A., Excitonic Couplings between Molecular Crystal Pairs by a Multistate Approximation. J. Chem. Phys. 2015, 142, 164107-164115. 69. Zheng, J.; Xie, Y.; Jiang, S.; Lan, Z., Ultrafast Nonadiabatic Dynamics of Singlet Fission: Quantum Dynamics with the Multilayer Multiconfigurational Time-Dependent Hartree (Ml-Mctdh) Method. J. Phys. Chem. C. 2016, 120, 1375-1389.

33



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

70. Busby, E.; Xia, J.; Wu, Q.; Low, J. Z.; Song, R.; Miller, J. R.; Zhu, X. Y.; Campos, Luis M.; Sfeir, M. Y., A Design Strategy for Intramolecular Singlet Fission Mediated by Charge-Transfer States In Donor–Acceptor Organic Materials. Nat Mater 2015, 14, 426-433. 71. Beljonne, D.; Yamagata, H.; Brédas, J. L.; Spano, F. C.; Olivier, Y., ChargeTransfer Excitations Steer the Davydov Splitting and Mediate Singlet Exciton Fission in Pentacene. Phys. Rev. Lett. 2013, 110, 226402-226406. 72. Tang, C. W.; VanSlyke, S. A., Organic Electroluminescent Diodes. Appl. Phys. Lett. 1987, 51, 913-915.

34


Page 34 of 35

Page 35 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC Graphic:

35


Functional Mode Singlet Fission Theory - The Journal of Physical

Recommend Documents