Is Vibrational Coherence a Byproduct of Singlet Exciton Fission? - The

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Is Vibrational Coherence a Byproduct of Singlet Exciton Fission? Marcin Andrzejak, Tomasz Skóra, and Piotr Petelenz J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b09124 • Publication Date (Web): 14 Dec 2018 Downloaded from http://pubs.acs.org on December 14, 2018

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The Journal of Physical Chemistry

Is Vibrational Coherence a Byproduct of Singlet Exciton Fission?

Marcin Andrzejak, Tomasz Skóra, Piotr Petelenz*

The K. Gumiński Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Gronostajowa 2, 30-387 Kraków, Poland

Abstract: A phenomenological wavefunction-based model of vibrationally coherent absorption modulation is proposed and applied to reproduce the triplet-triplet absorption spectra of bistriisopropylsilylethynyl (TIPS)-pentacene, with the objective of testing whether the singlet fission process in that system spontaneously generates coherent vibrational packets, as recently suggested for TIPS-tetracene. The model is parametrized by a complete set of Franck-Condon parameters obtained from methodologically consistent DFT calculations for all relevant normal modes in all relevant electronic states. The results strongly depend on inhomogeneous broadening of absorption bands, which is explicitly included. They very well agree with the recently published experimental coherence spectra of the pertinent system, validating our underlying principal assumption that the singlet fission process, which generates the observed triplet states, is neutral with respect to vibrational coherences. Experimental confirmation of this interpretational posit demonstrates that in the pentacene derivative, apparently in contrast to the case of its tetracene analogue, fission is not a source of vibrational coherence. Our finding suggests that although the singlet fission process may possibly in individual cases induce vibrational coherence, this feature is not a constitutive characteristic of the fission phenomenon. 1

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Introduction

Owing to applications-oriented interest in singlet exciton fission in acenes and their derivatives,1-6 the role of vibronic effects in this process is extensively studied using various experimental techniques. The target triplet-pair state lends itself to exploration, among others, by femtosecond four-wave mixing7 and pump-probe8 measurements, which reveal that in pentacene and 6,13bis(triisopropylsilylethynyl)-pentacene (TIPS-pentacene) the underlying radiationless transition conserves optically induced vibrational coherence. Recently a similar methodology has been used to investigate 5,12-bis(triisopropylsilylethynyl)-tetracene (TIPS-tetracene)9, which demonstrated that vibrational coherence in that system is conserved for most modes, but for one of them (moderately intense, at 760 cm-1) seems to be generated in the fission process as a kind of byproduct, attributed, according to some theoretical models,10 to vibronic effects (vide infra). This naturally opens the question whether TIPS-tetracene is special in this regard, or a similar mechanism is operative also in other systems, e.g. in TIPS-pentacene. In the present paper we attempt to answer this question by confrontation of experimental results available in the literature8 with our theoretical calculations. The experiment of Musser et al. which our calculations are meant to address was performed on a film of TIPS-pentacene. The sample was optically pumped by a 10 fs laser pulse, and probed by a 300 fs white pulse after a gradually increasing delay (up to 2000 fs). In the meantime, a third (dump) pulse was used to erase the signals due to coherent vibrational packets on the ground state energy surface. (Such three-pulse experiment is also known as populationcontrolled impulsive vibrational spectroscopy.11) In our present approach the dump signal will not be explicitly considered; instead, we will ignore the ground-state coherences it eliminated.

2


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The pump pulse generated a coherent superposition of vibrational states on the excited singlet (S1) energy surface, which presumably gradually propagated onto the triplet-pair (TT) surface, spanned by the lowest triplet states (T1) of the parent molecule and its nearest neighbor. The probe pulse monitored optical absorption leading to higher triplet states Tn. The temporal oscillations of this absorption exposed the vibrational wave packets that were coherently transmitted to the lowest triplet state from the initially excited singlet. As reported,8 the frequencies of these oscillations differed from those of the ground state normal modes, known from the Raman spectrum; some Raman-active modes were not detected at all. The amplitude of periodic changes in absorption, originating from a coherent vibrational wave packet, depends on the displacement of the equilibrium position between the initial and the final electronic state of the transition. This applies to the transition at the pumping (S1S0) as well as the probing (TnT1) stage. Therefore, it is conceivable that for some normal modes for which the coherent packets are formed in the S1 state and subsequently transmitted to the triplet manifold, their detection by triplet-triplet absorption is suppressed by the small differences between their equilibrium positions in different triplet states. In contrast, the contributions from originally weak coherences in other modes might be amplified and appear prominently if the shifts of the respective equilibrium positions upon excitations in the triplet manifold are sufficiently large. Our present objective is to calculate the pertinent displacements (and vibrational frequencies) by quantum chemical methods, in order to reproduce the experimental coherence spectra, and from potentially detected discrepancies with experiment to draw conclusions concerning the underlying physical mechanisms. Specifically, our aim is to detect putative excess

3



coherences hypothetically generated in the fission step, as assumed in Reference 9 for TIPStetracene. In the existing literature, the theoretical framework used to describe spectroscopic experiments of this kind is usually formulated in terms of quantum wave packets moving on a set of electronic energy surfaces, and is couched in the time-dependent density matrix formalism.12,13 Relatively little attention, though, is being paid to the manifold of higher intramolecular triplet states, since they are viewed as mere indicators of exciton flow from the initial singlet state to the final triplet-pair state. Our conjecture is that this might lead to misinterpetation of experimental signals (or lack of them) as dependent solely on the properties of the coherent wave packet on the S1/TT energy surfaces. Quantum chemical calculations for the triplet manifold are scarce, to some extent because, until recently, pertinent experimental evidence was in short supply. For pentacene, there is a reliable set of calculated triplet state energies and oscillator strengths for triplet-triplet transitions,14 unfortunately with no reference to the vibrational degrees of freedom that may be potentially involved in singlet exciton fission. Hence, our present paper is aimed to bridge the existing gap between the models of wave-packet dynamics and the normal-mode energetics in which these models are in fact rooted. Specifically, we attempt to reproduce the experimental spectra of time dependent photoinduced triplet-triplet absorption measured for TIPS-pentacene by Musser et al.. We make due allowance for the presence of the TIPS substituents although, in order to reduce the computational effort involved, we truncate them by replacing their isopropyl groups with hydrogen atoms, so that effectively the calculations are performed for 6,13-bissilylethynylpentacene (SE-pentacene). We believe that this model is still reasonably realistic, as the simplification affects only a part of the molecule which is rather distant from the chromophore, and alcane chains are pretty inert anyway. 4


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Quantum chemical calculations Within TURBOMOLE V6.4 2011 suite of quantum chemistry programs,15 we have used the TDDFT module with B3LYP energy functional and def2-TZVPP basis to calculate the energies of low-energy triplet-triplet transitions in the SE-pentacene molecule. In our calculations, the T1 state was treated as the ground state in the triplet manifold, and the higher triplet states were obtained in subsequent TDDFT procedure. In this context, the labelling of triplet states calls for a comment. In the following, we consistently conform to the notation of ref 8. It is based on the D2h symmetry group of the unsubstituted pentacene molecule, and the final states of two optical transitions from the lowest triplet state (T1=13B2u) that were observed by Musser et al. are referred to as T2=13B1g and T3=23B1g, respectively. One should bear in mind, though, that there are two more triplet states14 (23B2u and 13B3u) bracketed between T2 and T3, to which the transitions from T1 are forbidden by symmetry, so that in general count T3 is the fifth triplet state of the molecule. For the triplet states (T1, T2 and T3) involved in triplet-triplet absorption reported in ref 8 we have subsequently optimized the geometries and calculated the normal modes. Following earlier papers on similar subjects,16,17 for each transition of interest (T2T1 and T3T1) we have projected the geometry changes on the normal modes of the initial and final state. This was the crucial part of our calculations. The resultant displacements of equilibrium positions (dimensionless Franck-Condon parameters) for the modes that exhibit the strongest vibronic activity are collected in Table 1, and a complete list of FC parameters for all totally symmetric modes is accessible in Supporting Information.

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Table 1. Frequencies and Franck-Condon Parameters of Interpretationally Relevant Normal Modes for Low Energy Triplet-Triplet Transitions in the SE-Pentacene Molecule

dimensionless displacement wavenumber/cm-1 Normal mode

T1-T2 projected on T1-T3 projected on T1

T2

T3

T1

T2

T1

T3

1

233

238

235

-0.263

0.268

-0.164

0.167

2

267

267

267

-0.094

0.105

-0.032

0.033

3

617

616

611

-0.170

0.164

-0.015

0.021

4

642

645

639

0.073

-0.041

0.082

-0.077

5

717

722

720

-0.119

0.067

-0.030

0.002

6

811

803

803

0.216

-0.237

0.309

-0.305

7

941

941

939

-0.029

0.001

0.042

-0.042

8

950

950

950

-0.007

0.008

-0.003

0.003

9

1048 1051 1046 -0.007

0.000

-0.054

0.048

10

1119 1124 1127 -0.230

0.287

-0.094

0.131

11

1182 1183 1183 0.039

-0.008

-0.025

0.019

12

1239 1237 1240 -0.319

0.383

-0.303

0.225

13

1275 1264 1310 -0.629

0.858

-0.235

0.325

14

1431 1453 1425 0.807

-0.495

0.610

-0.601

15a

1374 1365 1393 -0.346

0.597

-0.266

0.266

16

1496 1506 1494 0.544

-0.097

0.297

-0.191

6


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17

1541 1552 1536 -0.049

0.289

-0.068

0.261

18

1598 1581 1572 0.381

-0.095

0.236

-0.198

a The

labeling of the modes is based on their energy sequence in the ground state.

For the sake of completeness, we have also calculated the normal modes for the ground and the first excited singlet state, which enabled us to evaluate the displacements of the equilibrium positions between the S1 and T1 state (possibly important for coherence transfer in the course of singlet fission) and between the ground (S0) and first triplet (T1) state (presumably relevant for the underlying intermolecular triplet exciton transfer). These results are shown in Table 2, along with the displacements for the S1S0 transition that shape the progressions in the initial pumping step.8

Figure 1. Schematic representation of electronic states considered in this paper for a model space of two normal coordinates. “SF” stands for “singlet fission”, represented by the wavy line.

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Figure 1 shows a scheme of the relevant electronic states and FC parameters involved, all projected on the idealized subspace of two normal modes. For the sake of transparency, the projections of the equilibrium position shifts on the axes corresponding to individual vibrational coordinates (qi and qj) are explicitly displayed only for one pair of electronic states; for other states they are obvious to figure out by analogy.

Table 2. Frequencies and Franck-Condon Parameters of Interpretationally Relevant Normal Modes for Low Energy Singlet-Singlet and Singlet-Triplet Transitions in the SE-Pentacene Molecule dimensionless displacement wavenumber/cm-1 Normal mode

S0-S1 projected on S1-T1 projected on S0-T1 projected on S0

S1

S0

S1

S1

T1

S0

T1

1

237

234

-0.359

0.376

0.013

-0.012

-0.339

0.366

2

270

268

0.529

-0.525

0.174

-0.176

0.699

-0.699

3

620

617

-0.207

0.217

0.059

-0.067

-0.156

0.158

4

653

639

0.088

-0.089

-0.037

0.029

0.054

-0.035

5

724

720

-0.195

0.192

0.085

-0.093

-0.114

0.080

6

800

809

0.304

-0.266

0.104

-0.099

0.415

-0.369

7

943

939

0.069

-0.058

-0.046

0.046

0.024

-0.019

8

951

948

-0.025

0.019

0.011

-0.010

-0.015

0.011

9

1022

1048

0.148

-0.071

0.027

-0.025

0.170

-0.107

10

1123

1126

-0.094

0.096

-0.061

0.066

-0.148

0.108

8


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11

1187

1183

0.378

-0.427

0.031

-0.015

0.426

-0.455

12

1228

1240

0.540

-0.511

0.152

-0.133

0.683

-0.648

13

1334

1326

0.146

-0.035

0.172

-0.186

0.409

-0.205

14

1420

1419

-0.627

0.537

-0.221

0.218

-0.716

0.753

15a

1438

1386

-0.182

0.449

0.135

-0.130

-0.071

0.250

16

1497

1505

0.075

-0.206

-0.175

0.189

-0.015

0.570

17

1552

1537

-0.435

0.436

-0.122

0.087

-0.647

0.459

18

1579

1583

0.309

-0.192

-0.044

0.031

0.211

-0.199

a

Note that the frequency of this mode considerably decreases upon excitation.

Comparison with the experimental results we are now interpreting8 suggests that the calculated frequencies conform to the usual several-percent standard of accuracy, with the familiar tendency for overestimation. When account is taken of the fact that the identities of some modes (recognizable by analyzing the atomic displacement patterns) change upon excitation, leading to interchanged frequency sequence in excited states, the observed frequency shifts are also very well reproduced. With due reservations concerning the accuracy of B3LYP calculations, worth noting is the perfectly predicted upward shift of 11 cm-1 for the interpretationally crucial ω6 mode (vide infra). As intuitively expected, the frequency and displacement characteristics of the S1 S0 and T1S0 transitions are qualitatively similar, with somewhat larger Franck-Condon parameters for the latter. The direct T1S1 reorganization energy turns out to be marginal (about 130-150 cm-1), which may facilitate the experimentally observed smooth coherence transfer.

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Theory Model. It should be emphasized that we presently strive not so much to directly unravel the mechanism of the singlet fission process itself, as to separate out the information obtainable from the coherences detected by triplet-triplet absorption, with emphasis on intrinsic limitations of this technique. For this reason, we circumvent the subtleties of optical coupling between different electronic

states,12,13

interaction

with

thermal

bath,10,18-20

etc.,

by

formulating

the

phenomenological part of our present treatment in simplistic wave function terms, where the main difference with respect to the standard static-absorption conditions consists in the fact that either the initial or the final state of the optical transition of interest is nonstationary.12 Our focus is on triplet-triplet absorption used to experimentally interrogate the fission-generated nonstationary wave packet on the energy surface of the first triplet state. In the present approach, only perfunctory attention will be given to the initial step of generating the primary packet on the singlet state energy surface, and to coherence transfer between the two energy surfaces involved. As our main target is the transient triplet-triplet absorption, we deliberately ignore the coherences generated in the ground electronic state, as their signatures were eliminated by the ingenious three-pulse pump-dump-probe protocol.8 Strictly speaking, for complete description of the fission process (TTS1) at least a dimer model is needed, since two molecules are indispensable to host the two emerging spin-entangled intramolecular triplets. However, the experiment8 addressed in the present paper is focused on intramolecular vibrations, and these are usually almost insensitive to the (weak) intermolecular interactions. As documented by the lack of Davydov-like vibrational splittings (or beats) in the observed coherent spectra, direct coupling between the normal modes of one molecule with those

10


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of its neighbor is evidently negligible, which precludes transfer of vibrational coherence from one molecule to another. This justifies the assumption that the coherences observed in triplet-triplet absorption originate from the energy surface of the same molecule at which the parent singlet exciton (along with the accompanying vibrational packet) was initially generated by the pump pulse. Hence, our model system is a single TIPS-pentacene molecule, its neighbor (not included explicitly in the model) serving merely as a receptacle for the second triplet exciton created in the fission process. Accordingly, in the following we explicitly consider the normal modes of one (effective) TIPS-pentacene molecule. The mechanism of the transition from the initial singlet excitation of the molecule under consideration to its triplet state will be viewed as a black box, of which the other moiety, maintaining the energy balance by accepting the other triplet exciton, is a part. Probing. The transient absorption measurement8 is readily described as conventional first-order absorption spectroscopy, its only peculiar feature consisting in the fact that the initial state of the transition of interest is nonstationary.12 It is prepared by the sequence of initial singlet-singlet pumping and the fission process. In building our simplified model, we implicitly rely on the known characteristics of the probing pulse,8 which is temporally rather long and spectrally white. In contrast to most previous treatments, here we express the wave packets in the basis of stationary harmonic-oscillator wave functions in the electronic states involved, which is best suited for our present purposes. Expressed in this basis, the vibrational wave packet in the T1 state assumes the form:     1  (t  0)  T1   x j exp  i j      | j ) , 2 j 0   

11


(1)


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where  is the frequency of the (th) normal mode involved, the kets | j ) denote the eigenstates of the corresponding vibrational Hamiltonian, and T1 stands for the electronic wave function. The values of the coefficients xj are a combined result of the wave packet shape generated in the initial singlet state by the pump pulse, and of its modulation during the transition onto the tripletpair energy surface (possibly, but not necessarily, through a conical intersection). The temporal variable  is the time elapsed between the pumping pulse (applied at =0) and the moment when the probing pulse is switched on (t=0). It sets the relative phases of the wave packet components at that moment and may be interpreted as a descriptor of the wave packet position on the potential energy surface. In the following, where we consider the interaction of the wave packet with the probing light wave, the phases controlled by  specify the initial condition. As derived in Supporting Information, for a monochromatic light wave of frequency  and electric field amplitude E0, the rate of light-induced TnT1 transition from the nonstationary state of equation 1 is given by:

I TT1n (  ; ; , t )  A



   ;  W k j

mjk

cosk  j   t    a mj    a mk ,

m , j ,k 0

where:

A

E 02

T1 M Tn

2

 Wmjk  Fmj Fmk x j x k 1  2 sin  k  j     2  k j  k  j    2

a mj  m  j   

T  T n

1





12


(2)

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with M denoting the dipole moment operator, and Fmj standing for the Franck-Condon overlap integrals between the j-th vibrational state on the T1 energy surface and the m-th vibrational state on the Tn energy surface.  T are the appropriate triplet state energies, and ηk-j is a damping factor accounting for finite time resolution  of the experiment (see Supporting Information); as mentioned by Musser et al., it suppresses primarily the contributions from higher frequency modes.8 The damping factor is obtained by averaging the absorption signal over the presumable inherent temporal uncertainty range of the experimental setup. Here this range is a fitting parameter, adjusted to best reproduce the pattern of observed relative normal mode intensities, shown in Figure S6 of the ref 8 Supplementary Information. It is the sole adjustable parameter invoked in our study, and its optimum value of 20.7 fs seems reasonable enough. In the energy domain, equation 2 still describes the absorption spectrum at maximum conceivable energy resolution, as decreed by the delta functions it contains. This resolution is attainable in no real experiment. Specifically, in the experiment of Musser et al. the vibronic structure is not resolved at all; the coherence spectra reported there (which we intend to reproduce) were integrated over a broad spectral range assigned to the respective electronic transitions (TnT1), and presented in the form of squared modules of the Fast Fourier Transform coefficients (|FFT|2) corresponding to individual normal modes. In order to mimic that procedure from the theoretical side, we have generated the integrated coherence output I Tintn    of a given mode by calculating the pertinent coherent amplitudes with full vibronic resolution provided by equation 2, and subsequently summing their squares over all vibronic transitions between the T1 and Tn vibrational manifolds (explicit formula may be found in Supporting Information). Equation 2 is a succinct analytical approximation of the TnT1 absorption, including the

13



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periodic time dependence that results from modulation by a coherent wave packet with normalmode frequency . As expected, the oscillating contribution to absorption intensity, being due to the nonstationary form of the wave packet in the T1 state, is governed by the products of the coefficients xj of different vibrational stationary states that contribute to that wave packet. It is crucial to note that the amplitude of the oscillations is also proportional to the product FmjFmk of the overlap integrals between the T1 and Tn vibrational eigenstates. This implies that the coherent contribution to the absorption spectrum can appear only if the vibrational eigenfunctions on the Tn energy surface are not orthogonal to those on T1, that is, if the corresponding oscillators are displaced with respect to each other (the frequency change being presently disregarded). This observation may be viewed as a kind of selection rule. The summation in equation 2 couples the terms FmjFmk, depending solely on the characteristics of the triplet manifold, with the products xjxk of the coefficients that specify the composition of the wave packet under study and are determined by the conditions of its preparation, namely, optical excitation to S1 followed by radiationless transition from S1 to T1. In order to decouple the two factors and approximately gauge the proclivity of triplet-triplet absorption to expose the coherent oscillations irrespective of their origin, for each mode  we define the descriptor (which hereafter will be referred to as triplet coherence detection factor)

J Tn    



F

Tn ,T1 0 ,i

F0T,in,T11 ,

(3)

i 0

separately for each of the two target triplet states. In contrast to normal practice for absorption transitions, the relevant vibrational overlap integrals that appear in equations (2) and (3) are calculated using the Franck-Condon parameters from columns 5 and 7 of Table 1, obtained by projecting the triplet-triplet geometry change on the normal modes of the (initial) T1 state.

14


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This is justified by the fact that just along this path the nonstationary oscillations drive the molecule on the T1 energy surface, which enforces the same trajectory to be probed in the target (Tn) electronic state. In the present context, the standard practice of projecting the shifts on the normal modes of the final state, implicitly taking into account the influence of Duschinsky rotation on the vibrational overlaps that govern the vibronic structure of the transition of interest, would not be warranted anyway. It is normally needed to correctly reproduce the intensity distribution between the different transitions within the Franck-Condon progression, whereas in the experimental results we are striving to reproduce the signal is integrated over the entire electronic transition, so that no vibronic resolution is retained. S1 pumping. In order to assess the role of the TTS1 radiationless transition in shaping the outgoing wave packet, it is necessary to know the form of its incoming counterpart, initially generated on the energy surface by the pumping pulse. With this aim, we again focus on a single normal mode. The pumping transition starts from the ground (electronic and vibrational) state; the composition of the final state is determined by the properties of the radiation pulse (different from those in the probing step). In view of the short pulse duration (10 fs), it is justified to approximate it as practically instantaneous. Its spectral shape (kindly provided by A. J. Musser) is adequately approximated by a spline-like function G(E) consisting of two halves of a Gaussian, shifted apart from each other, with the gap between them filled by a constant function (straight line), as displayed in Figure S1 of Supporting Information. The dispersion parameter  =500 cm-1 of the half-Gaussians is adjusted to best reproduce the actual steepness of the corresponding parts in the pump power spectrum, and their positions confine the pulse to the (experimental) energy range, set by the 500-

15



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650 nm wavelength limits, with the centroid at 2.18 eV (c = 17 593 cm-1) . This energy roughly corresponds to the lowest TIPS-pentacene excited singlet, accompanied by moderate vibrational excitation. Based on experimental results,7 for the S1 0-0 line we adopt the energy   S1 = 1.75 eV. (According to the same source, we locate the triplet-pair state at 1.72 eV.) In effect, the pulse generates on the S1 state energy surface a nonstationary coherent superposition of stationary harmonic oscillator wave functions (labeled by j, and corresponding to different oscillator energies), weighted by the coefficients c j  F jS,10, S0  G  S1  j   ,

(4)

(Their scale, set by a scaling factor for G(E), is arbitrary, but equal for all transitions under consideration, which guarantees consistency.) It follows that, in order to get simultaneously excited and form a coherent wave packet, at least two eigenstates characterized by different values of the quantum number j must simultaneously exhibit nonzero overlap with the initial ground state v=0 wave function. This is possible only for the modes with nonzero displacement of the equilibrium position between the ground and the excited singlet state (following a selection rule similar to that formulated earlier for triplet-triplet absorption). Ultimately, it is reasonable to expect large coherent oscillations for the modes endowed with large S1S0 Franck-Condon parameters. It should be mentioned that at this point we follow the standard practice for absorption transitions, calculating the relevant vibrational overlap integrals from the Franck-Condon parameters listed in column 5 of Table 2, obtained by projecting the S0-S1 geometry change on the normal modes of the (final) S1 state. In this case the path of the nonstationary  oscillations is set by the landscape of the S1 energy surface, so that now the trajectory argument concurs with the preference suggested by Duschinsky effect considerations.

16


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In analogy to J Tn (  ) invoked in the preceding section to characterize the coherences in triplet-triplet absorption, we now introduce a similar descriptor J S (  ) to describe the primary wave packets optically generated on the S1 energy surface in the pumping step 

J S (  )   c j c j 1 ,

(5)

j 0

where the coefficients cj are defined by equation 4. Accordingly, J S (  ) is governed by the conditions of primary wave packet preparation and by the properties of the singlet manifold (vibrational overlap integrals, determined by Franck-Condon parameters for the S1S0 transition), but does not depend at all on triplet manifold characteristics. It provides a crude measure of the supply of coherent oscillations entering the fission step. TTS1 fission. The nature of the TTS1 transition in acenes and their derivatives is now under continuous debate.7,8,10,18-27 After a period of vivid interest in the conical intersection hypothesis, preference is recently given to the role of vibronic coupling7,20,23,27-29 without explicitly invoking a conical intersection.

This distinction is beyond our present scope, since our approach

represents a purely kinematic description of the process, with no attempt to get dynamic insight into its mechanism (which is treated as black box). We presently consider this issue merely from the perspective of the coherences that are likely to propagate from the S1 to the T1 vibronic manifold of an individual molecule, where they could be monitored by triplet-triplet absorption spectroscopy.8 The modulation of TT absorption with which we have been dealing so far was that resulting from phase coherence directly induced by the pumping pulse in the vibrational wave functions on the S1 PES. The emergent nonstationary wave packets further propagating

17



(presumably in unchanged form) onto the T1 PES were shown to cause temporal oscillations of the TnT1 vibrational overlaps (Cf Supporting Information), manifested in periodic changes of TT absorption intensity. One can speculate that,9 should the transition between the S1 and T1 energy surfaces occur abruptly (i.e., within a few femtoseconds), it could act as a sudden perturbation, generating (by virtue of the time-energy uncertainty principle) secondary coherent wave packets in the vibrational manifold, just as the pump pulse does. In the present case this is unlikely, since the pertinent electronic transition is known to take a longer time, on the order of 50-100 fs, perceptibly exceeding the oscillation period.9 Nonetheless, the hypothetical extra coherences, if generated, would contribute to the amplitude of the measured periodic absorption changes. Conceivably, the periodic modulation of TT absorption might have a still different component. According to some vibronic models,10,20 the TTS1 transition may be driven by one of the vibrational modes, which then acts as the reaction coordinate, its oscillations tipping a part of the excited state population from one PES to the other. [In our model this would amount to periodic modulation of the coefficients xj that appear in equation (2).] In that case, the intensity of triplet-triplet absorption would inevitably follow the resultant periodic changes of T1 population. Although mediated by a different mechanism (affecting the triplet population, not the phase of the vibrational wave packet), these absorption changes would be phenomenologically indistinguishable from those described earlier, and would contribute on equal footing to the overall amplitude of probe absorption modulation, as apparently assumed in reference 9. If the TIPS-tetracene case fulfills the conditions necessary for population oscillations predicted by Berkelbach, these contributions may be the source of the excess coherence reported by Stern et al..

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In our calculations, we consistently ignore these hypothetical contributions caused by the electronic TTS1 transition as described above. We include solely the coherent contributions directly generated by the initial pumping pulse (xj=cj), so that a putative coherence deficit in the calculated spectra may be attributed to the involvement of the fission process. Accordingly, we assume in the way of a working hypothesis that the transition from the S1 to the T1 vibronic manifold occurs smoothly in the sense that the degrees of vibrational excitation (embodied in the v quantum numbers) do not change. The marginal energy surplus (from experiment known to be about 0.03 eV for TIPS-pentacene 7) is presumably absorbed by the lowfrequency phonon bath which includes vibrations of the surroundings, the inter-moiety vibrations, and potentially also incoherent intramolecular vibrational excitations, especially the lowfrequency ones (ω1=233 cm-1, ω2=267 cm-1) which for technical reasons are not observed in the experiment under consideration,8 but perfectly fit the pertinent energy gap. The above picture is consistent with the calculated shifts of equilibrium positions between the S1 and T1 energy surfaces, which in experimentally observable modes are rather small. In this way we effectively assume that the form of the T1 vibrational wave packet is a copy of its S1 counterpart, i.e. that the fission process is perfectly neutral as far as vibrational coherences are concerned; deviations of our computed results from experimental observations will permit to pinpoint the shortcomings of this simplistic posit. In fact, when described in terms of a complete vibronic approach, 27-29 individual vibronic eigenstates are bound to be combinations of respective vibrational factors multiplied by the S1 and T1 electronic wave functions. The small energy gap between the S1 and triplet-pair energy minima, combined with (mostly small) Franck-Condon parameters, suggest that the strongest

19



mixing should occur just between the states with the same v number, which lends likelihood to the above crude picture. For the sake of completeness, it should be noted that, owing to the mixing between the S1 and the TT state, mediated by charge transfer configurations and approximately described in terms of the superexchange concept,5,10,18,19 the dimer eigenstate of TT parentage, carrying no intrinsic intensity from the ground state, inevitably borrows some transition dipole moment from the S1S0 transition. In effect, a coherent wave packet may be instantaneously formed on the TT energy surface directly by the initial pumping pulse, to be joined later by the one propagating from the S1 PES. Although in the experiment of Musser et al. the short-time signals are strongly affected by a coherent artifact, Figure 2b of their paper seems to exhibit in the T2T1 spectral range some genuine coherent modulation. The modulation is very weak (just what as expected for this mechanism), and consistently neglected here, since in this paper we are focusing on the leading contributions. Sample texture. Until now, we have been consistently considering the TIPS-pentacene film (prepared and examined by Musser et al.) as a set of identical molecules with equal S0→S1 transition energies of 1.75 eV.7 There are good reasons to deem this picture oversimplified. A pretty obvious one is the width of absorption bands in the TIPS-pentacene steady-state absorption spectrum reported by Musser et al. Spectral decomposition of the recorded signal reveals the Gaussian dispersion of the lowest-energy peak to be ca. 750 cm-1. In similar spectra measured for an individual TIPS-pentacene domain,30,31 the corresponding band is significantly narrower, with Gaussian dispersion of about 550 cm-1, which roughly agrees with the value of ca. 450 cm-1, observed for the pentacene single crystal.32 In fact, there is no doubt that structural perfection of a small crystallite is inferior to that of a single crystal, and the low-energy degrees

20


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of freedom contributed by the flexible TIPS substituents are bound to increase the net spectral width, fully justifying the excess of 100 cm-1 for TIPS-pentacene. As the measurements of Musser et al. were performed on an (inevitably disordered) thin film, the larger width of the spectrum they registered is readily attributed to inhomogeneous broadening that results from averaging over an ensemble of domains characterized by different S0→S1 transition energies. Accordingly, based on the familiar sum rule, the Gaussian dispersion for the first-band excitation energy distribution in the sample of ref 8 should amount to about 500 cm-1 (since 5002 + 5502 ≈ 7502). The averaging over an ensemble of domains with different excitation energies also explains the noticeable shift between the first absorption maximum in steady-state absorption spectrum and the maximum of ground state bleach in the transient absorption map.8 Let us consider a Gaussian distribution of S0→S1 transition energies, centered around 1.75 eV. The dominant fraction of the TIPS-pentacene population has their transition energies in close vicinity of 1.75 eV, so the resultant steady-state absorption maximum is located there. In contrast, the energy of the pump pulse used for femtosecond experiments (covering the range between 1.86 and 2.48 eV, cf . Supporting Information) is considerably higher. Consequently, the pump favors excitation of the domains with blue-shifted absorption spectrum, and these domains, despite being a minority, absorb an inordinately large part of pump radiation with respect to the majority of crystallite population. This imbalance suppresses the relative population of unexcited domains with blue-shifted transition energies, so that in effect the GSB maximum drifts to shorter wavelengths. The magnitude of that shift explicitly depends on the dispersion of the Gaussian distribution governing the incidence of different excitation energies in the population, and may be estimated as follows.

21



The contribution A of an individual domain (characterized by a given 0-0 transition energy =S1–S0) to absorption of the pump pulse may be gauged by summing the squared vibrational wavepacket coefficients of equation (4) over all contributing vibrational states in all contributing normal modes. As stipulated by the equation, the result A=A(–ħc) depends on the location of the domain’s 0-0 absorption line with respect to the pump emission band. With the distribution of 0-0 transition energies sampled at a discrete set of points, the total pump-induced ensemble-averaged absorption may be approximated by the sum of domain contributions A(–ħc) weighted by the population density D() of the corresponding 0-0 transition energies. The GSB maximum occurs at the energy for which the product D() A(–ħc) is maximal.

As implicitly assumed earlier, the pertinent distribution D() (representing the

probability density of finding a molecule with a specific transition energy) is approximated as Gaussian. Direct calculations (based on the Franck-Condon parameters presented earlier) indicate that for ensembles with Gaussian dispersion of 400-450 cm-1 the resultant GSB shift is close to that observed on the transient absorption maps of Musser et al. (maximum at 686 nm, vs the observed 690 nm). Those dispersion values are not far from 500 cm-1, estimated above by comparing the absorption bandwidth observed for the film and for a single TIPS-pentacene domain, the discrepancy being well within the limits set by low resolution of the absorption spectra and by Gaussian approximation for the band shape. This consistency supports the estimated scale of inhomogeneous broadening, and promotes it from the status of a justified conjecture to that of an established attribute of the sample on which the experiments were performed. In the vibrational coherence context, a slight bias for the lower-side estimate of 400450 cm-1 is justified by the fact that the latter is rooted in experimental evidence concerning

22


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ground state bleaching, collected by using the same pump pulse as for the vibrational coherence study, and hence affected by the same set of inherent experimental errors. Accordingly, the vibronic coherence spectrum of the sample is generated following a protocol similar to that used above for ground state bleaching. Using equation (S20) of Supporting Information, the domain coherence spectra I Tintn    are calculated for a set of relative 0-0 line positions with respect to the pumping band, and subsequently summed, with weights given by Gaussian distribution with dispersion   450 cm-1. For future reference (vide infra), analogous procedure is used to get the ensemble-averaged values of coherence supply descriptors J S (  ) defined by equation 5.

The resultant spectra are to be discussed in the next section. At this point, however, it is worthwhile to note that the influence of inhomogeneous broadening on relative coherence amplitudes in different modes turns out to be crucial. The most dramatic effect is registered for the 811 cm-1 vibration (6) which would be practically unobservable, were the sample a homogeneous set of TIPS-pentacene molecules with the nominal S0→S1 transition energy of 1.75 eV. In that case, spectral overlap between the pumping band (starting at 1.86 eV) and the 0-1 and 0-2 lines in this mode would be negligible, whereas higher vibronic replicas, being endowed with negligible absorption intensity on account of the modest Franck-Condon parameter, would not appear at all.

Results Calculated coherence flow. There is little doubt that the propyl groups of TIPS-pentacene barely influence the low-energy electronic excited states engaging primarily the -electron core of the molecule, and that, vice versa, the propyl vibrations are practically unaffected by this kind of

23



Page 24 of 43

excitations. Consequently, in the present context the TIPS-pentacene vibronic characteristics may be adequately represented by those of SE-pentacene. Based on this assumption, in our model of vibrational coherence flow described in the foregoing sections, for the requisite Franck-Condon parameters we adopt those calculated for the latter molecule. Table 3 shows for each vibrational mode  the main features of the vibrational coherence flow, namely the coherence supply factor J S (  ) , defined by the conditions of optical excitation, the triplet coherence detection factors J Tn    , characterizing the proclivity of the molecule’s triplet manifold to translate the coherence of the wave packet on the T1 PES into modulation of triplet-to-triplet absorption (for both transitions under study), the product of the two pertinent factors for each of the TnT1 transitions (which approximates the relative amplitudes of TnT1 modulation), the damping factor  determined by the temporal resolution of the experimental setup, and the exact amplitude I Tintn    (in arbitrary units), calculated according to equation (S20) of Supporting Information. This last quantity corresponds to (|FFT|2) of ref 8.

Table 3. Descriptors of Coherence Flow Calculated for TIPS-Pentacene

normal cm-1

JS a

J T2

J T3

J S J T2

24

J S J T3




I Tint2

a

I Tint3

a

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mode 1

233

0.009

0.184

0.116

0.002

0.001

0.967

0.168

0.074

2

267

0.016

0.066

0.023

0.001

0.000

0.957

0.060

0.008

3

617

0.016

0.120

0.011

0.002

0.000

0.784

0.044

0.000

4

642

0.005

0.052

0.058

0.000

0.000

0.767

0.001

0.002

5

717

0.018

0.084

0.021

0.002

0.000

0.715

0.017

0.001

6

811

0.045

0.152

0.215

0.007

0.010

0.645

0.116

0.245

7

941

0.006

0.021

0.030

0.000

0.000

0.541

0.000

0.000

8

950

0.001

0.005

0.002

0.000

0.000

0.534

0.000

0.000

9

1048

0.010

0.005

0.038

0.000

0.000

0.453

0.000

0.000

10

1119

0.016

0.161

0.066

0.003

0.001

0.394

0.007

0.001

11

1182

0.501

0.028

0.018

0.014

0.009

0.342

0.010

0.004

12

1239

1.000

0.222

0.211

0.222

0.211

0.295

1.000

1.000

13

1275

0.006

0.420

0.165

0.002

0.001

0.266

0.002

0.000

14

1431

1.717

0.521

0.409

0.894

0.702

0.144

0.916

0.876

15

1374

0.894

0.240

0.186

0.215

0.166

0.187

0.294

0.206

16

1496

0.136

0.369

0.207

0.050

0.028

0.097

0.009

0.004

17

1541

1.107

0.035

0.048

0.038

0.053

0.066

0.001

0.001

18

1598

0.135

0.264

0.166

0.036

0.022

0.028

0.000

0.000

25



aJ

S

is arbitrarily normalized to unity for the ω12 mode which yields the most intense triplet-

triplet modulation, and so are the final amplitudes of columns 9 and 10. The JT values (columns 4 and 5) are those directly obtained from equation (3), without any further normalization.

At the first glance, the most striking feature of the results is the high-frequency cutoff due to the damping factor that ensues from the finite temporal resolution of the experimental setup, as already reported by Musser et al.. This feature apart, it is evident that the modes that induce major modulation of triplet absorption (ω12=1239 cm-1, ω15=1374 cm-1 and ω14=1431 cm-1) are mostly those combining substantial singlet coherence supply with large triplet coherence detection factors, both descriptors being limited by the corresponding Franck-Condon parameters. However, the ω6=811 cm-1 mode compensates its marginal supply in the singlet manifold (caused by the low frequency, limiting the spectral overlap with the pump pulse) by substantial FC shifts of the equilibrium position upon both triplet excitations, so that it ultimately appears in triplet-triplet absorption with moderate amplitude. A similar effect, although less evident because of a much lower overall scale, is noticeable for ω3=617 cm-1, which (despite marginal pumping efficiency), is predicted to contribute a nonnegligible signal to T2T1 coherence, notably with a vanishing T3T1 counterpart. In contrast, the ω11=1182 cm-1 vibration, which can be pumped with reasonably high efficiency owing to its appreciable FC parameter for singlet excitation, has its triplet coherence output curtailed to negligible values because of the small FC parameters for the corresponding electronic transitions. Comparison with experiment. In Figures 2 and 3 the calculated coherent output I Tintn    is compared with relative intensities of the experimentally observed oscillating contributions to

26


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triplet-triplet absorption, as reported in the supplementary Figure S6 of ref 8. The comparison should be taken with a grain of salt, since (evidently in consequence of the demanding multistep procedure of experimental data collection and numerical processing), the resultant experimental signals to be interpreted inevitably exhibit considerable background noise, especially in the (weaker) T2T1 coherences.

Figure 2. Integrated coherence output IintT2( ) (sticks) versus experimental vibrational coherence spectrum (continuous line) for the T2T1 transition (from the Supplement of ref 8). The frequencies of experimentally observed coherences are displayed above the corresponding peaks.

As it has already been mentioned, the vibrational frequencies are consistently overestimated; this is an inherent feature of the DFT method we have used, and of the routinely adopted harmonic approximation. At the same time, it is evident that the qualitative features of the mode intensity pattern in triplet modulation spectra are very well reproduced, namely: 1. The modes that appear prominently in observed triplet coherences are correctly indicated.

27



2. In perfect agreement with experiment,8 despite its substantial Raman intensity the ω11 mode (predicted in the ground state at 1187 cm-1 and observed at 1158 cm-1), is not anticipated among the triplet coherences (because the relevant FC parameters are negligible). The band at that frequency in the T2T1 spectrum is ignored by Musser et al., being an artifact of the numerical procedure eliminating the ground-state vibrational coherence, which, as they state, cannot be done neatly. 3. The predicted signals of some modes (such as ω3 mentioned above, and ω5), not listed explicitly in ref 8 as coherent modulators, can be found among the wiggles in the noisy T2T1 coherence spectrum. Actually, in view of the above-mentioned several-percent error margin in their frequencies, an attempt to assign them to specific features of the experimental spectrum would be premature, account taken of the indubitably high noise. Yet, the fact is that some appreciable peaks do appear just in the low frequency part of the T2T1, where they are anticipated.

Figure 3. Integrated coherence output IintT3( ) (sticks) versus experimental vibrational coherence spectrum (continuous line) for the T3T1 transition (from ref 8). The frequencies of experimentally observed coherences are displayed above the corresponding peaks.

28


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Quantitatively, the signals at about 1400 cm-1 seem at the first glance to be somewhat overrated by the calculations, but this impression results mostly from the stick representation of the simulated spectra; when allowance is made for the differences in experimental band widths, the calculated intensities are quite close to the mark. Especially worth noting in our results, though, is the good agreement registered for the ω6=811 cm-1 mode; the main interpretational point to be taken later is based on the fact that for this mode no intensity deficit is detected. Summarizing, even at quantitative level the agreement may be considered excellent. Looking into the underlying computations, it should be borne in mind that for triplet states (which are crucial here) the quantum chemical methods we have employed are less well tested, and a poorer reproduction of experimental data would be excusable; yet, their rendering seems entirely plausible. By and large, account taken of the fact that our results pertain to several independent electronic energy surfaces, most of them representing excited states (for which quantum chemistry methods are always less reliable), the agreement with experiment may be considered remarkably good. Moreover, it should be noted that spectral intensities are always more sensitive to shortcomings of the model than energies. Photoinduced absorption (PIA) we are addressing here is effectively a nonlinear optical phenomenon; hence, the coherence amplitudes monitored by PIA in the triplet manifold are bound to be particularly sensitive to inaccuracies of the quantum chemistry methods employed, since they are obtained as products of the singlet supply part and the triplet coherence detection part, where both factors result from independent calculations so that the errors might potentially cumulate. With this view, the overall agreement exceeds our expectations.

29



Role of the singlet fission process. As hypothesized recently,9 transition of the model dimer from the S1 energy surface may spontaneously generate a coherent vibrational wave packet on the PIA of the product triplet(s). Since in our calculations this effect is entirely disregarded, then if it would indeed occur in TIPS-pentacene, the simulated spectra we have generated, when compared to the experimental ones,8 should exhibit some deficit in coherence amplitudes. The phenomenon described above was presumably observed for TIPS-tetracene9 which is a homologue of the system we are presently investigating. The vibration with frequency of 760 cm-1, causing moderately intense modulation of triplet-triplet absorption that was gradually appearing in the course of singlet fission, was interpreted as a product mode of that process.10,20 By analogy, one would expect that in our case the mode that is the most likely to exhibit similar behavior would be either ω5 (717 cm-1) or ω6 (811 cm-1). The former is not observed in experimental T3T1 absorption, and in T2T1 is barely detectable (if at all, in view of the background noise). This leaves ω6 as the sole candidate, and its observed coherent signal is perfectly accounted for without invoking any coherence production in the fission process. Admittedly, our treatment predicts only relative contributions of different modes, so it would be in principle conceivable that the intensities that are really underestimated by the calculations are those of all other modes, but this seems utterly improbable, especially in view of the experimental evidence for TIPS-tetracene, pointing at a mode around 800 cm-1.

Discussion The experience we have gathered in course of the present project inspires the following reflection:

30


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Coherent activity of a normal mode in photoinduced absorption is governed by several, usually independent, factors. In the first place, the method of excitation must be appropriate for generating a coherent wave packet. In practice, this implies that the exciting stimulus (usually a pumping laser pulse) must be short enough (in time domain), and must be tuned to one of electronic states of the system being excited. Of course, the system must be capable of absorbing the exciting light, i.e. the corresponding transition must carry a nonnegligible transition moment. Moreover, the vibrational mode to be coherently excited must exhibit a nonnegligible displacement of the equilibrium position (Franck-Condon parameter) between the initial and final electronic state involved in the pumping transition. Once the wave packet is generated (irrespective of the way this was accomplished, be it by direct optical transition, or by its combination with an unknown black-box mechanism inherent to singlet fission), its coherent modulation of the PIA spectrum is also subject to obvious selection rules: the probing laser pulse must be tuned to one of the transitions that are allowed from the generated electronic excited state (which is the initial state of the probing transition), and the vibrational mode in which the wave packet was generated must exhibit a nonzero shift of the equilibrium position between the initial and the final electronic state of the probing transition. As originally pointed out by Musser et al., the ultimate amplitude of the coherent contribution to absorption is also limited by finite temporal resolution of the probing pulse, which introduces a mode-dependent damping factor resulting from averaging over the intrinsic time error of the experiment. For experimentally feasible values of the latter the damping affects primarily high-frequency modes. Within these general rules, the ability of a given normal vibration to coherently modulate optical absorption is governed by its Franck-Condon parameters for the specific electronic excitation in hand; generally, the same mode may be active in this regard for one electronic 31



transition, but mute for another. The main novelty our results bring is that this is not only possible in principle, but that these differences do occur for some modes, and that they may indeed be substantial. This is just what we have noted for the TIPS-pentacene ω11=1187 cm-1 mode (ground state frequency), for which the shift of the equilibrium position upon S1S0 excitation is substantial (-0.427), but those for TnT1 transitions turn out to be negligible (0.039 for T2T1 and -0.025 for T3T1). In contrast, the 13 vibration (1334 cm-1 in S0), for the S1S0 transition exhibiting a negligible parameter of -0.035, is extremely prone to coherent excitation from the T1 energy surface, with the FC parameters of -0.629 and -0.235 for the T2T1 and T3T1 transitions, respectively, i.e. an order of magnitude larger. Admittedly, this boost does not suffice to make the resultant coherence observable. In the recent experiments8,9 the behavior of individual vibrational modes in the fission process is probed by monitoring their coherent signatures in triplet-triplet absorption. Our present results demonstrate that, while detection of such a coherent contribution from a given mode is a definite proof that a coherent wave packet in this mode penetrates from the singlet to the triplet manifold or is generated in fission, the actual amplitude of triplet absorption modulation is no direct gauge of the degree to which this mode is promoting fission, or, should it be a product thereof, what is its yield. In order to draw any conclusions concerning the latter two issues, the observed coherence amplitudes would have to be scaled by the TnT1 FC factors, and only then different modes could be compared with each other. Thus, the need to know the FC parameter pattern for specific modes and specific electronic transitions is an inherent limitation of inferences based on PIA coherence analysis.

Conclusion

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For TIPS-pentacene we know the pertinent pattern now. Our calculations are based on this pattern and on the proviso that the fission process neither suppresses nor promotes vibrational coherence. The agreement of the calculated coherence amplitudes with experiment supports the underlying posit. It is very unlikely that the inherent errors of the present approach would be accidentally compensated to that extent. It is equally unlikely that putative discrepancies would be systematically masked by some hypothetical mechanism of coherence generation or attenuation. Definitely, the singlet-pumping step entirely accounts for the coherences observed in triplet-triplet absorption. In our opinion, this is a strong indication in favor of the hypothesis that in TIPS-pentacene, in contrast to TIPS-tetracene,9 singlet fission does transmit coherent vibrational excitations to the product TT state, but is not their source. Consequently, with the insight our present calculations provide, specifically for the TIPSpentacene case we can now firmly assess that only the modes having large singlet coherence supply exhibit large oscillating signals in TnT1 absorption, and their calculated intensities plausibly reproduce those experimentally observed. Hence, in contrast to the TIPS-tetracene case, there is no indication of a spontaneous coherence-generating mechanism at the fission stage. If for TIPS-tetracene this mechanism is viewed as black box, for TIPS-pentacene the corresponding box seems to be empty. Taking tentatively for granted (based on ref 9) that such a mechanism does exist, it apparently is not a general feature of fission in all systems.

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Associated Content Supporting Information

Complete tables of Franck-Condon parameters for low-energy singlet and triplet excitations, derivation of the optical transition rate from a nonstationary state, derivation of the damping factor, derivation of the integrated coherence output, analytic approximation of the pump emission spectrum (PDF). This material is available free of charge via the Internet at http://pubs.acs.org.

Author Information

Corresponding Author *[email protected]

Notes The authors declare no competing financial interest.

Acknowledgements The authors express their gratitude to Andrew Musser for providing the emission spectrum of the pump used in the experiments of ref 8, and for useful comments. This research was performed with equipment purchased owing to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (contract no. POIG.02.01.00-12-023/08).

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Figure 1. Schematic representation of electronic states considered in this paper for a model space of two normal coordinates. “SF” stands for “singlet fission”, represented by the wavy line. 84x63mm (300 x 300 DPI)


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Figure 2. Integrated coherence output IintT2( ) (sticks) versus experimental vibrational coherence spectrum (continuous line) for the T2T1 transition (from the Supplement of ref 8). The frequencies of experimentally observed coherences are displayed above the corresponding peaks.



Figure 3. Integrated coherence output IintT3( ) (sticks) versus experimental vibrational coherence spectrum (continuous line) for the T3T1 transition (from ref 8). The frequencies of experimentally observed coherences are displayed above the corresponding peaks.


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Is Vibrational Coherence a Byproduct of Singlet Exciton Fission? - The

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