Theory of fluorescence spectrum dynamics and its application to determining the relaxation characteristics of the solvent and intramolecular vibrations

Journal Pre-proof Theory of fluorescence spectrum dynamics and its application to determining the relaxation characteristics of the solvent and intramolecular vibrations Roman G. Fedunov, Igor P. Yermolenko, Alexey E. Nazarov, Anatoly I. Ivanov, Arnulf Rosspeintner, Gonzalo Angulo PII:

S0167-7322(19)34903-7

DOI:

https://doi.org/10.1016/j.molliq.2019.112016

Reference:

MOLLIQ 112016

To appear in:

Journal of Molecular Liquids

Received Date: 6 September 2019 Revised Date:

18 October 2019

Accepted Date: 24 October 2019

Please cite this article as: R.G. Fedunov, I.P. Yermolenko, A.E. Nazarov, A.I. Ivanov, A. Rosspeintner, G. Angulo, Theory of fluorescence spectrum dynamics and its application to determining the relaxation characteristics of the solvent and intramolecular vibrations, Journal of Molecular Liquids (2019), doi: https://doi.org/10.1016/j.molliq.2019.112016. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Theory of fluorescence spectrum dynamics and its application to determining the relaxation characteristics of the solvent and intramolecular vibrations Roman G. Fedunova , Igor P. Yermolenkob , Alexey E. Nazarovb , Anatoly I. Ivanovb,∗, Arnulf Rosspeintnerc , Gonzalo Angulod a Voevodsky

Institute of Chemical Kinetics and Combustion, 3 Institutskaya Str., Novosibirsk, 630090, Russia b Volgograd State University, University Avenue 100, Volgograd, 400062, Russia c Department of Physical Chemistry, University of Geneva, 30 quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland d Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland

Abstract A general analytical expression for the transient fluorescence spectrum is derived. The formation of a wave packet in the excited state of a fluorophore is described, assuming that the pump pulse has a Gaussian time-profile. The expression explicitly connects the relaxation characteristics of the medium with the spectral dynamics of a fluorophore. Fitting the expression to experimental spectral dynamics allows obtaining the solvent relaxation function. So far this approach was applicable for the analysis of experimental data when the pump pulse does not populate excited sublevels of intramolecular high-frequency vibrational modes. Here, the approach is generalized to include vibrational relaxation in the excited electronic state. In this case, fitting to the experimental spectral dynamics provides reliable information not only on the solvent relaxation, but also on the relaxation time constants of intramolecular high-frequency vibrational modes. This approach is applied to the excited state dynamics of coumarin 153 in multiple solvents, obtained from broadband fluorescence upconversion spectroscopy. Keywords: microscopic liquid dynamics, broadband fluorescence upconversion spectroscopy, time dependent Stokes shift, vibrational relaxation, wave packet dynamics

1

1. Introduction

25 26

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Ultrafast photochemical reactions initiated by a short 27 laser pulse proceed on the timescale of solvent relaxation.[1– 28 15] Immediately after photoexcitation the chromophore is 29 in the Franck-Condon state embedded in a non-equilibrium 30 environment. The instant rate constant of chemical trans31 formation depends on the state of the environment and, 32 hence, varies during the course of its relaxation. The dise33 quilibrium of the nuclear subsystem means that the kinetic 34 regime has not yet been reached, and the rate constant de35 pends on time.[16, 17] The dynamics of ultrafast reactions 36 can strongly depend on the magnitude of the solvent re37 laxation time constants. Thus, to simulate ultrafast chem38 ical dynamics, precise knowledge of the solvent relaxation 39 characteristics is necessary and mandatory. 40 The formal theory of time-resolved spectroscopy is well 41 developed[18–22] and has been successfully applied to the 42 simulation of spectral dynamics in real systems.[23–26] 43 The theory is well adapted for solving the direct prob44 lem of modelling spectral dynamics with known input pa45 rameters. However, determining the solvent relaxation 46 parameters from spectral dynamics requires tackling the 47 inverse problem, which is solved by analysing the time48 resolved fluorescence spectra. Usually the form of the 49

∗ Corresponding

author. E-mail: [email protected]

Preprint submitted to Elsevier

50

time-dependent fluorescence band is only used to improve the accuracy of determining the position of its maximum or center frequency (first moment)[27–29] and the evolution of its width and asymmetry are not used when determining the solvent relaxation function. It is clear, that such an approach of modelling the spectral dynamics, makes it difficult to separate the effects of intramolecular vibrational relaxation and solvent relaxation in the Stokes shift dynamics. As a consequence, fitting without such a separation may lead to incorrect solvent relaxation characteristics. In a recent paper it was shown that the temporal evolution of the fluorescence maximum is relatively independent of the presence or absence of vibrational relaxation, while the higher spectral moments are indeed dependent.[30] Another, crucial problem is related to the uncertainty of the initial position of the fluorescence band.[27, 29] In a typical fitting the dynamic Stokes-shift data starting at a certain time, t0 , is available. Thus the time-zero spectrum is needed. A widespread and popular approach for its estimation is based on the steady-state spectra in an apolar reference solvent.[27] Thus, there is necessarily an unresolvable uncertainty for these short-time data. The time-zero fluorescence band is determined by the spectral characteristics of the pump pulse and any successful theory has to include an explicit description of the excitation stage. The simultaneous modelling of the shape and posiOctober 18, 2019

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

tion dynamics of the time-resolved fluorescence spectra is108 expected to significantly improve the accuracy and relia-109 bility of the obtained solvent relaxation parameters. 110 Ultrafast spectral and chemical dynamics in condensed111 phases are governed by intramolecular vibrational modes112 of the solutes as well as the solvent relaxation modes.113 While the effect of solvent relaxation on the spectral and114 chemical dynamics is typically incorporated in the theo-115 retical description, intramolecular vibrational relaxation116 is rarely accounted for.[31–37] The problem of relaxation117 of quantum vibrational modes is usually solved within the118 model of the quantum Brownian oscillator, which is the119 simplest model to describe open quantum systems coupled120 to a thermal bath of quantum oscillators.[18, 38–49] An in-121 fluence functional path-integral method of Feynman and122 Vernon is the main approach to construct the quantum123 master equation of the Brownian oscillator motion.[50, 51]124 In this approach, the total Hamiltonian is divided into two125 parts: a system and a heat bath, which is characterised by the interaction bath spectral density function.[50] The key 126 quantity here is the reduced density operator describing the system evolution for the collective reaction coordinate127 experiencing a friction due to interaction with the heat128 bath. 129 In terms of the quantum Langevin equation the spec-130 tral density is related to the time-dependent friction kernel131 function and to the stochastic force operator.[50, 52, 53]132 In order to simulate the dissipative dynamics of the open133 quantum system, it is necessary to solve the stochastic dif-134 ferential equation.[54] These approaches also follow from135 the method of Feynman. As pointed out in Ref. 49, Stock-136 burger and Grabert derived a stochastic Liouville equation137 for the reduced density matrix using the influence func-138 tional with Gaussian coloured noise.[55] Non-Markovian139 dissipative dynamics have been treated with both quan-140 tum and semiclassical formalisms.[29, 56–63] In this article, we have derived an expression for the time resolved fluorescence spectra (TRFS) using perturbation theory for the interaction of a pump pulse with a fluorophore. The obtained expression connects the transient fluorescence spectra directly with the static and dynamic characteristics of both fluorophore and solvent, as well as with the parameters of the pump pulse. The key point of the approach is the idea of a “reaction coordinate”. It is of primary importance that the distribution of the imaging points along this reaction coordinate determines the TRFS completely. Starting with the well-known141 spin-boson model, the distribution function in the excited142 electronic state is integrated over all nuclear coordinates,143 except for the reaction coordinate. This expression con-144 tains in explicit form the relaxation function of the low145 frequency modes of both solvent and solute. This relax-146 ation function is assumed to be arbitrary, meaning that147 the expression can be fitted to the TRFS in solvents with148 Debye or non-Debye relaxation, including Gaussian and149 Brownian relaxation. It should be stressed, that here all150 effects of friction, including memory, which are discussed151 2

in any description of the motion along the reaction coordinate in terms of a Langevin equation, are accounted for in terms of a nuclear relaxation function. The general expression for the TRFS is significantly simplified in the limit of a strong electron-vibrational interaction, which provides tremendous advantages when fitting to experimental spectral dynamics. The approach is generalised to additionally include the relaxation of high-frequency vibrational modes of the solute. In the TRFS simulations, changes of the frequencies of these intramolecular vibrations are also accounted for. The aim of this paper is (i) to derive a general expression for the transient fluorescence signal, (ii) to investigate the effect of vibrational relaxation on transient fluorescence spectra, and (iii) to apply this approach to broadband time-resolved fluorescence spectra of coumarin 153 – a well-studied benchmark system – in a series of solvents. 2. Theory We present a theoretical instrument that establishes a direct connection of an arbitrary (solvent) relaxation function and an arbitrary intramolecular vibrational relaxation function with the spectral dynamics, which is measured in time-dependent fluorescence experiments. (for a full exposition see the supporting information). Fitting spectral dynamics obtained with this instrument to the observed spectral dynamics does not require any additional assumptions, in particular, the initial position of the fluorescence spectra. It is important that the spectral evolution during excitation is taken into account since the pump pulse duration is often comparable with the timescale of fast relaxation modes. Thus, we are introduced only final expressions for the fluorescence spectrum. The time-resolved fluorescence spectrum can be written in following form X I(ω, t) ∼ ω 3 F (~n, m)Φ ~ 1 (ω, t, ~n, m), ~ (1) {~ n,m} ~

where F (~n, m) ~ =

M Y γ=1

p f

(f) nγ , mγ , ~Ω(i) γ , ~Ωγ ,

2Ervγ (g)

! ,

(2)

~Ωγ

is the Franck-Condon factor [64, 65] with ~n = (n1 , n2 , . . . , nγ , . . . , nM ), ~ is the Planck constant, M is the total number of quantum vibrational modes active in the electronic transition considered, Ervγ and nγ are reorganization energy for transition from the ground (g) to excited state (e) and quantum number of the γth vibrational mode, re(x) spectively, Ωγ is the vibrational frequency of the initial (i) and final (f) state of the γth vibration, the last term in the parenthesis represents the change of the oscillator equilibrium position along the oscillator coordinate and f is provided in the SI.

152

Here, the quantity Φ1 has the following form

Z

t

Φ1 (ω, t, ~n, m) ~ = −∞

154 155 156 157 158

153

dξP (~ω + ∆G + Er −

where t is the current time of spectral evolution initi-159 ated by a pump pulse. The zero moment is defined as160 the moment of passage of the maximum pulse through the161 dye solution. Thus, t is synchronized with the time delay162 between the pump and the fluorescence detection, which163

X

mγ ~Ω(e) γ +

X

γ

nγ ~Ω(g) ~ γ , t, ξ, m),

(3)

γ

is variable. The rest quantities are ω is the fluorescence frequency, ∆G is the free energy gap between the ground and excited states, Er is the reorganization energy of the medium and P is the solution of the master equation describing the decay of the high-frequency modes

   M  X mγ − 1 1 1 dP (y1 , t, ξ, m) ~ mγ + 1 0 00 P (y1 , t, ξ, m ~ γ) + P (y1 , t, ξ, m ~ γ ) −mγ + d P (y1 , t, ξ, m) ~ +Ze−1 ϕ(y1 , t, ξ, m) ~ = d u u dξ τ τ τ τ vγ vγ vγ vγ γ=1 (4) where the initial condition is

168 169

P (y1 , t → −∞, ξ, m) ~ →0 164 165 166 167

(5)170

tional transition rate constants of down and up transitions, correspondingly, for γth oscillator with mγ quanta. In the u d thermal state, τvγ and τvγ are connected by the equation (e)

The vector m ~ = (m1 , m2 , ..., mγ , ..., mM ) differs from171 0 the m ~ γ = (m1 , m2 , ..., mγ + 1, ..., mM ) and m ~ 00γ = (m1 , m2 ,172 ..., mγ − 1, ..., mM ) by only an additional quantum in the173 u d are the vibra-174 and mγ /τvγ γth vibrational mode, mγ /τvγ

d u = exp(~Ωγ /kB T ) where kB and T are Boltzmann /τvγ τvγ constant and temperature, respectively. Here, the distribution function ϕ along the reaction coordinate y1 in a given vibrational state m ~ is defined as follows

( ) 2 F (~0, m) ~ y˜12 (t − ξ) 2ξ 2 [~δωe (t − ξ, ~n)] ϕ(y1 , t, ξ, m) ~ = exp − − 2 − σe (t − ξ) 4Er kB T τe 2σe2 (t − ξ)

y˜1 (t) = y1 − 2Er Q(t)

~δωe (t, m) ~ = ~ωe + ∆G − Er −

(7) X γ

mγ ~Ω(e) γ

− y˜1 (t)Q(t)


σe2 (t) = 2Er kB T 1 − Q2 (t) + ~2 τe−2  h i2  s P (e)     ~ω + ∆G − E − m ~Ω 2 X γ e r γ γ 4π Er kB T ~ F ( 0, m) ~ exp − Ze = τe   4Er kB T + 2~2 τe−2 {m} 4Er kB T + 2~2 τe−2   ~ Q(t) =

175 176 177 178 179 180 181 182 183 184 185

1 πEr

Z 0

∞

(6)

J(ω) cos ωt dω ω

(8) (9) (10)

(11)

where ωe is the carrier frequency of the excitation pulse186 and τe is its duration, Q(t) is the normalized relaxation187 function of the classical solvents modes with spectral den-188 sity J(ω), Ze−1 is the normalization factor which guaran-189 tees that the population of the excited state is equal to190 unity after the pump pulse passes. 191 To clarify the meaning of the functions P (y1 , t, ξ, m), ~ 192 we first note that in the absence of vibrational relaxation,193 1/τvγ = 0, P (y1 , t, t, m) ~ is the time dependent distribution194 function along the reaction coordinate y1 in a separate vi-195 brational state m. ~ This quantity is obtained by integrating196 3

eq (4) over ξ up to t. The quantity Ze−1 ϕ(y1 , t, ξ, m)∆ξ ~ is a part of the population distribution on the mth ~ vibrational sublevel of the electronic excited state at time t which was created by the pump pulse at time ξ during the interval ∆ξ. In other words, this summand not only properly describes the particle creation in the excited state but it also describes the propagation of the wave packet (the particle distribution) on the excited state surface during the time interval from ξ to t. Third, the first two summands in eq. (4) describe the kinetics of the irreversible transitions between vibrational sublevels starting with the

197 198 199 200 201 202 203 204 205

appearance of the particles in the excited electronic state. Since the free energies of states with different sets of the vibrational numbers m ~ differ only by a constant, the laws of motion along the reaction coordinate are the same for all vibrational states. As a consequence, the vibrational transitions do not affect the dynamics of the particle distribution along the reaction coordinate in the excited electronic state, and eq. (4) properly describes the particle distribu206 tion dynamics when vibrational relaxation proceeds.

In the case when the pump pulse populates the electronic excited state with its intramolecular vibrational modes in the ground states, the signal of the time-resolved fluorescence can be considerably simplified X I(ω, t) ∼ ω 3 F (~n, ~0)Φ2 (ω, t, ~n) (12) {~ n}

where

207

(

t

2

[~δωe (t − ξ, ~n)] ~2 ω ˜ 2 (t − ξ, ~n) 2ξ 2 − 2 − Φ2 (ω, t, ~n) = − ξ) exp − 2E k T τ 2σe2 (t − ξ) r B −∞ e X ~˜ ω (t, ~n) = ~ω + Er + ∆G + nγ ~Ω(g) γ − 2Er Q(t) Ze−1

Z

dξσe−1 (t

) (13) (14)

γ

"

#

~δωe (t, ~n) = ~ωe + ∆G − Er − ~ω + Er + ∆G +

208 209 210

211

212 213 214 215 216 217

218 219 220 221 222 223 224

225 226 227 228 229 230 231 232 233 234 235 236

X

nγ ~Ω(g) γ

Q(t)

(15)

γ

These equations provide a method of calculating the dy-237 described in Ref. 68.1 In brief, sum frequency generation of namics of the florescence spectrum for an arbitrary time238 the fluorescence was performed using 1340 nm pulses, ex239 hibiting a pulse front tilt matched to the fluorescence-gate delay. 240 angle of 21◦ , in a 100 µm thick BBO crystal (EKSMA, 241 θ = 40◦ , φ = 0◦ ). The upconverted fluorescence was 3. Experimental 242 relayed to a home-built grating- spectrograph (Newport, 243 53006BK01-010R) via a fibre-bundle and detected using 3.1. Chemicals 244 a cooled CCD camera (Andor, iDus DV420A-BU). The Coumarin 153 (C153, CAS 53518-18-6), BBOT (CAS 245 photometric response was determined using a set of sec7128-64-5), coumarin 6H (C6H, CAS 58336-35-9) and DCM 246 ondary emissive standards, which have been calibrated (CAS 51325-91-8) were used as received. All solvents (see 247 on the steady-state fluorescence spectrometer. The wavethe SI for purity and supplier information) were used as 248 length calibration of the home-built spectrograph was perreceived. 249 formed scanning the calibrated wavelength output of a 250 Cary Eclipse fluorimeter (coupled via the fibre of the FLUPS) 3.2. Steady-State Spectroscopy 251 and a Hg-lamp. The chirp on the fluorescence signal was Absorption spectra were recorded on a Cary 50 UV-252 determined using the instantaneous response of a sample Vis spectrophotometer, the wavelength accuracy of which253 of BBOT in the same solvent as that of the C153 samwas calibrated using a Holmium oxide glass standard.[66]254 ple and correspondingly corrected for, before further data Fluorescence spectra were recorded on a Fluoro-Max 4, the255 analysis. All experimental spectra were corrected for the photometric response curve of which has been determined256 optical chirp and the photometric response and converted using a set of secondary emissive standards.[67] 257 to be presented with an x-axis proportional to energy (see 258 Ref. 69) before comparison with the fitted spectra. 3.3. Time-Resolved Spectroscopy Measurements were performed in commercial cuvettes259 3.4. Data Treatment (Starna) with an optical pathlength of 1 mm and quartz In order to make the comparison of experimental and windows of 1.25 mm thickness each. The optical density simulated spectra simpler we opted for single value descripof the samples amounted to between 0.2 and 0.3 at and tors of the spectral information, i.e. the spectral moments beyond the excitation wavelength of 400 nm. The samples were bubbled with argon during the measurement in order 1 Given the fact, that the time-resolution over the entire spectral to continuously refresh the excitation volume. Broadband FLuorescence UPconversion Spectra (FLUPS), observation range is not known, global fitting approaches such as the one in Ref. [30] were not considered. with a time-resolution of 170 fs (judged from the FWHM of the intense, spectrally integrated, upconverted Raman signal at approx. 3000 cm−1 ) were recorded using a set-up 4

291

4. Discussion

314 315

292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313

The model outlined in section 2 (see also the SI) in-316 cludes a rather large number of parameters determining317 the spectral dynamics of molecules excited by a short laser 318 pulse. However, these parameters can be divided into two 319 groups: A) (Energy) parameters that are relevant for the description of the steady-state electronic spectra (Tables 2320 and 3) and B) (relaxation) parameters that describe the dynamics of both the solvent (Table 4) and intramolec-321 ular vibrations. The first set of parameters can be de-322 termined from fitting the simulated spectra to experimen-323 tal absorption and steady-state fluorescence spectra (cf.324 Section 4.1). Keeping those fixed, the relaxation param-325 eters are then determined from fitting the spectrodynam-326 ical model to the experimental time-resolved fluorescence327 spectra (cf. Section 4.2). Such a division of the fitting328 into two stages provides important advantages, since at329 each stage the number of fitting parameters is reduced to330 331 a minimum.[29] Another advantage of this approach is the possibility of332 an explicit description of the formation of a wave packet333 on the free energy surface of an excited electronic state334 with known energy parameters. The relaxation parame-335

ters determine the wave packet evolution that starts simultaneously with its formation. Thus, in contrast to the approach commonly pursued,[27] there is no uncertainty in determining the initial position of the wave packet. This is of paramount importance for the precision of determining the characteristics of the fastest relaxation modes. 4.1. Energy characteristics of coumarin 153 The steady state spectra contain exclusively information about the energetic characteristics of the transitions since they are associated with the stationary states of the solute in each solvent. While always true for absorption, in the case of fluorescence this is strictly the case as long as the relaxation dynamics proceed at time scales much shorter than the fluorescence decay, such that the non-equilibrium fluorescence emission contributes negligibly to the time-integrated fluorescence signal. For the sake of robustness and to reduce the range of solutions a joint nonlinear fitting of the steady-state absorption, A(ω) and fluorescence spectra, I(ω) is exploited. The transition probabilities are calculated within the Golden Rule approximation, given by the following form[74, 75]

 h  i2  P     ~ω + ∆G − Er + γ mγ ~Ω(e) γ ~ , A(ω, ∞) ∝ ω F (0, m) ~ exp −   4Er kB T   m ~  h  i2  P (g)     ~ω + ∆G + Er + γ nγ ~Ωγ X I(ω, ∞) ∝ ω 3 F (~n, ~0) exp −   4Er kB T   ~ n X

5

(17a)

(17b)

(g)

336 337 338 339 340 341 342 343 344 345 346

tions of C153.[78, 79] This set of vibrational modes, presented in Table 2, provides excellent fits to the steady-state absorption and fluorescence spectra. This is very important for the evaluation accuracy of the solvent relaxation parameters from fitting to time-resolved fluorescence spectra that evolve towards stationary ones. Fig. 1 shows the fits to experimental spectra at three representative solvent polarities (all other spectra are shown in the SI). Comparison of the experimental and simulated steady-state spectra shows not only excellent agreement upon visual inspection (Fig. 1), but also when comparing the spectral moments (see Fig. 2).

√

m2,fit / meV

A(ω)/ω, I (ω)/ω 3 norm.

347

(e)

where Ωγ , Ωγ are vibrational frequency of the ground362 and excited states, respectively. These expressions in-363 clude all the necessary energy parameters of the problem,364 namely: the free energy change ∆G, the reorganization365 energy Er of low-frequency modes and the reorganization366 energy Ervγ and frequency Ωγ of the high-frequency in-367 tramolecular vibrational modes, respectively. The reorga-368 nization energies Ervγ are contained in the Franck-Condon369 370 factors F (~0, m) ~ and F (~n, ~0). It should be noted that eq (1) reduces to the fluo-371 rescence spectrum expression of eq (17b) in the limit of372 373 t → ∞.

√

hω ¯ Figure 1: Experimental steady-state absorption (blue) and fluorescence spectra (red) of coumarin 153 in dimethylsulfoxide (DS), diethyl ether (EE) and 2-methylbutane (MB), together with simula-374 tions (dashed lines) using the parameters from Tables 2 and 3. The375 discrepancy between fit and experiment in the absorption spectrum 376 at high energies is due to a higher electronic transition. 377 378

348 349 350 351 352 353 354 355 356 357 358 359 360 361

In order to further reduce the ambiguity in the ad379 justable parameter sets the following strategy was applied: 380 In low polar solvents both steady-state spectra exhibit a 381 pronounced vibronic structure, which favors the determi382 nation of the relevant intramolecular vibrational frequen383 cies and their reorganization energies with maximum pre384 cision. In fact, we determined these parameters from fits 385 to the steady-state spectra of C153 in methylbutane. The 386 ensuing parameter values are listed in Table 2 and were 387 considered solvent independent. In particular, a set of 388 three vibrational high-frequency modes was necessary, i.e. 389 γmax = 3. All of the six high frequency modes (three for 390 absorption and their slightly altered excited state analogs) 391 are close to previously identified3 high-frequency vibra392 393

3 Ω(g)

= 680, 1240, 1560 cm−1 ; Ω(e) = 510, 1230, 1590 cm−1 .

394

6

m2f,exp / meV

Figure 2: Comparison of fluorescence spectral moments from the experimental spectra and the multimodal fits to them (lower panels) as well as the difference between the two (res, in upper panels). Solvents with larger dielectric constant than tetrahydrofuran have been excluded, as it was not possible to fully record the spectra of coumarin 153 on the fluorimeter used.

The parameters ∆G, Er , and Erv , obtained from the fitting, are collected in Table 3. One can see that, as expected, the reorganization energy associated with the high frequency modes, Erv , varies only marginally with solvent polarity. At the same time, the alterations of the free energy gap, ∆G, and the reorganization energy associated with low-frequency modes, Erv , are fairly large and systematic. In particular, the free energy gap decreases and the reorganization energy increases upon increasing the solvent polarity. Figure 3 demonstrates the excellent correlation of these changes with solvent polarity, which is in line with previous findings for C153.[27] It is however important to point out, that particularly the low-frequency reorganization energy is offset by 33 meV from the values previously determined.[80] In fact, this difference in Er is a consequence of the fact that the current model - contrary to Ref. [27], where the intramolecular relaxation is assumed to be infinitely fast - also accounts for the Stokes shift in nonpolar solvents, where by definition dielectric solvation was assumed to be zero. This is important, as it implies that ”solvation dynamics” will always be contam-

Table 1: Solvent properties and steady-state spectral characteristics of coumarin 153 (at 20 ◦ C). ε is the dielectric constant and n the refractive index of the solvent (these data were either taken directly from Ref. 76 or recalculated to 20 ◦ C from data in Ref. 77). In addition the parameters of the time-zero analysis in the framework of the pure dielectric relaxation approach (PDRA) are given. The corresponding spectra in all solvents are shown in the supporting information.

solvent

solvent properties  n

spectral characteristicsa √ √ m1a m2a m1f m2f (eV) (meV) (eV) (meV)

2-methylbutane (MB) cyclohexane (CX) di-n-butyl ether (BE) di-i -propyl ether (IP) diethyl ether (EE) butyl acetate (BA) benzyl acetate (BZ) propyl acetate (PA) ethyl acetate (EA) tetrahydrofuran (TH) butyronitrile (BN) acetonitrile (AC) ethylene glycol (EG) dimethylsulfoxide (DS)

1.83 2.02 3.08 3.88 4.20 5.01 5.70 6.00 6.02 7.58 24.83 35.94 38.69 47.01

3.229 3.196 3.148 3.138 3.131 3.085 3.041 3.078 3.080 3.066 3.024 3.021 2.928 2.966

1.3509 1.4270 1.3968 1.3655 1.3495 1.3918 1.5020 1.3830 1.3698 1.4050 1.3820 1.3410 1.4323 1.4783

200 199 204 204 206 211 209 214 214 210 213 215 213 216

2.664 2.637 2.520 2.499 2.467 2.362 2.326 2.324 2.332 2.342 2.256 2.216 2.112 2.165

time-zero-analysisb δ0 Γinh mPDRA (0) 1f (meV) (meV) (eV)

200 199 203 202 201 204 209 213 214 208 208 209 198 208

32 80 88 97 143 187 150 149 163 204 207 298 263

2 95 109 132 163 155 174 178 174 191 208 227 213

∆m1f c (meV)

2.632 2.583 2.575 2.566 2.521 2.477 2.513 2.514 2.500 2.460 2.457 2.365 2.401

a

-5 63 76 99 158 151 189 182 158 204 241 253 236

m1x first moments and the square root of the corresponding second spectral moments, m2x of the steady-state absorption and fluorescence spectra. The data were obtained from the fits of eq (17) and parameters from Tables 2 and 3 to the experimental spectra. b PDRA parameters: spectral shift, δ , and inhomogeneous broadening, Γ 0 inh , and first spectral moment of the estimated time-zero emission specc The difference ∆m PDRA (0) − m , which is the estimate for the trum, mPDRA (0), using the formalism given in section III of Ref. 27. 1f = m1f 1f 1f shift of the first spectral moment of fluorescence due to solvation dynamics, following Ref. 27.

Table 2: The Franck-Condon active frequencies in the ground state, (g) (e) (g) Ωγ , their ratio between excited and ground state, αγ = Ωγ /Ωγ , and the associated relative reorganization energies yγ = Ervγ /Erv , (with Erv tabulated in Table 3) for coumarin 153. The parameters were obtained by fitting eq (17) to the experimental steady-state absorption and fluorescence spectra in 2-methylbutane.

395 396

397

(g)

γ

Ωγ (cm−1 )

αγ

yγ

1 2 3

726 1161 1605

0.800 1.028 1.061

0.305 0.373 0.322

Table 3: The best fit parameters (∆G, Er and Erv ) obtained from the stationary absorption and fluorescence spectra of coumarin 153 in a series of solvents.

inated by these, intramolecular low frequency relaxation, dynamics. 4.2. Relaxation Dynamics Having determined the energy characteristics, one can begin to search for the relaxation parameters of the solvent and intramolecular vibrations using nonlinear fitting of the simulated signal eq (1) to experimental fluorescence upconversion spectra for several time delays. Here the relaxation function based on the current approach, Q(t), is approximated by a multiexponential function Q(t) =

N X

xi e−t/τi ,

(18)

i=1

where xi and τi are the weight and time constant of the P solvent relaxation modes, i xi = 1. Eq. (18) follows from 7

solvent

−∆G (eV)

Er (meV)

Erv (meV)

MB CX BE IP EE BA BZ PA EA TH BN AC EG DS

2.952 2.924 2.840 2.825 2.805 2.730 2.690 2.708 2.714 2.710 2.646 2.625 2.524 2.571

33 31 68 74 86 111 112 123 117 114 137 155 174 151

256 253 251 251 252 256 250 259 261 253 252 252 240 255

I (ω)/ω 3

hω ¯ Figure 4: Selected experimental (colored dots) time-resolved fluorescence spectra of coumarin 153 in dimethylsulfoxide and the corresponding fitted spectral reconstructions (dashed black lines). The sharp features at short times, marked with an asterisk, represent Raman contributions from the solvent. The gap at approx. 1.85 eV is necessary to omit residual signal due to the second harmonic of the gate. The spectral reconstructions are calculated with the best fit parameters listed in Table 4.

Figure 3: Dependence of the free energy gap, ∆G, the reorganization energy of the low frequency modes, Er , and that of the high frequency intramolecular modes, Erv on the solvent polarity function f () − f (n2 ) (◦), where f (x) is defined as (x − 1)/(x + 2). For the first two quantities, the data are compared to those from Ref. 81 (•).

398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420

421 422 423

where xi Er is the reorganization energy of the ith relaxation mode. It should be emphasized that in the general case the developed theory operates with an arbitrary relaxation function Q(t), i.e. in eq (18) the sum is not limited to exponentials but can include, for example, Gaussian or Brownian oscillatory modes. For the fitting of the dynamics of the time-resolved fluorescence spectra, a set of time delays is chosen to provide a uniform distribution of the peak positions of the spectra between their initial and final position (see the SI for details). Figure 4 depicts experimental and simulated fluorescence spectra of C153 in dimethylsulfoxide at representative time delays. One can see that the blue edge of the simulated spectra deviates from the experimental one at short (< 0.2 ps) time delays. The experimental excitation carrier frequency of about 3.1 eV considerably exceeds the free energy gap, |∆G|, leading to the excitation of high-frequency intramolecular vibrations. These excited vibrational states and their relaxation are most likely to be responsible for the spectral dynamics424 in the blue edge. In fact, the ideal initial fluorescence425 spectrum necessarily has to extend to 3.1 eV. However,426 there are some experimental limitations: 427 1. For most of the solvents studied we have used a428 420 nm long-pass filter in emission, thus suppressing429 430 any fluorescence above 2.95 eV. 8

I (ω)/ω 3

the general expression, eq (11), when the spectral density J(ω) is presented by the sum of several Debye relaxation spectra X 2ωτi , (19) J(ω) = xi Er 1 + ω 2 τi2 i

hω ¯ Figure 5: Selected experimental (colored dots) time-resolved fluorescence spectra of coumarin 153 in dimethylsulfoxide and the corresponding simulated spectral reconstructions assuming three different vibrational relaxation times. The solvent relaxation parameters are the same as in Figure 4.

2. Fluorescence reabsorption (and thus loss) by the 1 mm thick chromophore solution itself is unavoidable and we have estimated the ensuing attenuation of signals in the wavenumber range of absorption to be at maximum on the order of 30%. 3. The limited time-resolution of approximately 170 fs will blur out any signals which decay with times sig-

431

nificantly shorter than this.

484 485

432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483

molecule.[71, 83] In the calculations we assume the intramolecular vibrational relaxation to proceed irreversibly and with equal time constants for all vibrational modes that ultimately leads to populating only their ground states. This is done in order to reduce the number of adjustable parameters. The difference between theoretical and experimental data on the second moment could be reduced if this approximation was removed, but this would require introducing a large roto-vibrational spectrum of lower frequency modes and the associated rates, in other words, more sophisticated mechanism of redistribution/relaxation of intramolecular vibrational modes. We believe that the observed discrepancies in the second moments show that the time-dependent fluorescence spectra contain information about vibrational relaxation, which can be extracted using a more complex theory that describes vibrational relaxation in more detail.5

Experiments performed without the long-pass filter al486 low for assessing the influence of these individual contri487 butions.4 In the absence of the long-pass filter still essen488 tially no emission close to the excitation energy can be 489 observed. It thus seems like the most important reason 490 for the non-observation of entirely unrelaxed emission is 491 the finite time-resolution. The latter has multiple contri492 butions, such as the excitation with a pulse of finite pulse 493 duration (ca. 100 fs), the ensuing group velocity mismatch 494 in the sample cuvette between pump and fluorescence, or 495 the sum frequency generation with the gate pulse of finite 496 pulse duration (ca. 100 fs). Only the first of these three 497 effects has been taken into account in our calculations. 498 We thus have to conclude that important information on 499 the early spectral dynamics is significantly distorted or 500 obscured. Fitting such distorted spectra can lead to the underestimation of the vibrational relaxation timescales 501 4.3. Comparison with Previous Approaches and we have to settle for not being able to accurately determine them with sufficient precision. As a consequence502 4.3.1. The Pure Dielectric Relaxation Approach (PDRA) There are two major differences between the current we have decided to fix the vibrational relaxation time at503 504 approach and the pure dielectric relaxation approach.[27, the level of 50 fs for the fitting, which is in line with pre505 85] First, in the classical PDRA the contribution from vious observations for C153.[27] Figure 5 shows the effect 506 vibrationally excited states is set apart. Moreover, the of overestimating the vibrational time constant, which can 507 time-zero spectrum is calculated based on the stationary result in broadening the simulated spectra in the blue in 508 emission observed in low polar solvents. In the PDRA, the time window from 0 to 1 ps. It is still possible that the relaxation time is longer up to 100 fs, as the longer time509 all observed relaxation is dielectric and external to the spectra are well reproduced by this time. The short-time510 fluorophore, which is considered merely as a beacon. In spectra are then better reproduced, though, by using 50 fs511 the current approach the time zero spectrum is calculated which may be a compensation for the short wavelength512 from the known energetics of the system taking into ac513 count the excess excitation energy, and it therefore evolves problems pointed above. Another way to compare the experimental spectra with514 also following vibrational relaxation. Besides, the second the simulations consists in analyzing the spectral moments.515 major difference is that the rest of the relaxation is not Figure 6 shows that the dynamics of the first spectral mo-516 uniquely ascribed to the solvent but reflects all low frement are well reproduced in the entire time domain, with517 quency contributions including those of the fluorophore, the exception of cyclohexane, di-ethyl ether and acetoni-518 making it no longer an innocent reporter.[86] Therefore trile (the latter two are shown in the SI), which essen-519 the PDRA assumes that the fluorescence bandshape is intially show ”overshooting” dynamics.[82] Already in ref. 27520 variant during relaxation (this is in principle only true in 521 −1 the Maroncelli and co-workers obtained a clear narrowing (300 cm ) transition dipole moment representation). It is clear −1 and a more subtle blue shift (150 cm ) of the C153 spec-522 that whatever dynamics are involved in accounting for the trum in cyclohexane that takes place on a 10 ps time scale.523 intrinsic Stokes shift in e.g. entirely nonpolar solvents, are Also in the work of Ernsting and Sajadi[82] this effect has524 not accounted for.[87] Within the published analyses following the PDRA, been observed, discussed in more detail and was given varoriginated by Maroncelli and Fleming, we have to take into ious explanations. The fact, that the overshooting is seen account that not all groups follow exactly the same recipe. in some solvents and not in others is related to the solvent This concerns which quantity is analyzed (maximum or relaxation times and its amplitude and therefore is better first moment of the emission band) and how this quantity observed the faster the former and the smaller the latter, becoming apparent in cyclohexane and acetonitrile. The dynamics of the second spectral moments are only 5 Other work [71] has attributed all additional dynamics to narpoorly reproduced. This is not surprising, as the changes rowing by cooling rather than vibrational redistribution. However, we think that the interpretation of the results in that particular in the second moment are supposed to monitor kinetics of work could be masked by other effects, as the broadening in the redistribution/relaxation of intramolecular Franck-Condon spectra could be due to structural distributions.[84] A clear indicaactive vibrations and ensuing vibrational cooling of the tion of this is the loss of mirror symmetry when going from spectra at short times to the stationary one. In fact, cooling should go from non-mirror-symmetric spectra to mirror-symmetric ones, while for quadratic coupling just the opposite is expected, as observed in ref. 84.

4 Here the pump was set perpendicular to the gate pulse to suppress Rayleigh scattering,

9

m2f / meV √ m1f (t) − m1f (∞) / meV Figure 6: Representative experimental (black dots) and simulated (red dashed line) time-dependencies of the first and second spectral moment of the emission of C153. Solvent abbreviations are given in Table 1.

is analyzed (with or without convolution, for example). For the sake of consistency with the key considerations in Ref. 27 we proceeded as follows. We use the first spectral moment of the spectra in the transition dipole moment representation as the most unbiased single value descriptor (compared to e.g. the spectral maximum). Rather than fitting the entire time-range we exclude those data which experience serious distortions caused by the finite duration of the instrument response function (i.e. t < 300 fs). However, we fix the spectral position at time zero to the one estimated using the considerations of Ref. 27, namely that no relevant relaxation takes place in nonpolar solvents. In practice the fitting function is as follows, m1f (t) = ∆m1f Q(t) + m1f (∞), 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542

Table 4: Parameters for Q(t) with the model developed in this paper.

solvent

(20)

where ∆m1f is the difference between the first spectral moment of the calculated time-zero spectrum, m1f (0), and that of the steady-state emission, m1f and the relaxation function, Q(t), is defined in eq. (18). Note that the steady543 state value is a very good estimate for the time infinity 544 emission spectra (as assumed above), as a comparison of 545 the two values in Table 5 shows. 546 Fig. 7 presents a comparison of the relaxation func547 tions obtained with the above two approaches for C153 in 548 tetrahydrofuran and dimethylsulfoxide. One can see that 549 the approach developed here leads to a much faster ini550 tial decay of the relaxation function. The reason for the 551 differences between the relaxation functions given by the 552 two approaches is mainly due to the difference in the po553 sition of the time-zero spectra. This is well seen in Fig. 7. 554 Although the difference between the time-zero positions is 555 only about 1% (in dimethylsulfoxide), this results in con556 siderable differences in the solvent relaxation parameters. 10

CX BE IP EE BA BZ PA EA TH BN AC EG DS

x1

x2

1.00 0.66 0.66 0.53 0.52 0.52 0.41 0.41 0.55 0.54 0.81 0.45 0.56

0.34 0.34 0.47 0.33 0.26 0.36 0.32 0.45 0.41 0.19 0.19 0.30

x3

0.15 0.22 0.23 0.27 0.05 0.35 0.14

τ1 (ps)

τ2 (ps)

0.01 0.04 0.12 0.05 0.14 0.09 0.08 0.08 0.17 0.17 0.13 0.10 0.16

3.5 5.6 0.89 2.7 2.3 0.92 0.53 1.5 1.4 1.0 4.0 1.5

τ3 (ps)

19 27 18 3.1 18 36 7.3

Except for technical details (like the accounting or not for vibrational relaxation) the main difference between the present approach and the previously published approaches is that no reference measurement in a nonpolar solvent is required to access the entire dynamics. In the previous approximations the Stokes shift in low polar solvents is ascribed to the distortion and displacement of the excited state with respect to the ground state and therefore the relaxation in these solvents is connected to the internal coordinates of the solute. Such an opinion is based on the presumption that the solvent relaxation is purely connected to dielectric relaxation or, in other simpler terms, to the vibrations / rotations / librations of the dipoles of the solvent. Thence, if the solvent is nonpolar, by defini-

Table 5: Parameters for Q(t) with the PDRA according to Maroncelli and Fleming using eq. (20).[27, 85]

solvent BE IP EE BA BZ PA EA TH BN AC EG DS

∆m1f a (meV) 64 78 102 168 163 200 194 171 223 263 289 264

x1

x2

x3

x4

τ1 (ps)

τ2 (ps)

τ3 (ps)

τ4 (ps)

0.01 -0.13 0.03 0.08 0.29 0.44 0.40 0.09 0.21 0.61 0.21 0.37

0.52 0.41 0.58 0.33 0.31 0.31 0.53 0.51 0.48 1.86 0.21 0.44

0.32 0.72 1.92 0.43 0.40 0.21 0.07 0.41 0.31 -1.46 0.46 0.19

0.16

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0.25 0.42 0.34 0.30 0.76 1.1 1.5 0.49 0.53 0.86 1.2 1.01

6.5 5.0 2.9 2.8 11.4 5.6 107 2.1 2.6 1.0 21 7.4

48

-1.51 0.18 0.04

-0.01 0.12

3.5 23 368

20 144

m1f (∞)b (eV) 2.517 2.500 2.463 2.362 2.333 2.343 2.330 2.339 2.261 2.223 2.104 2.163

m1f c ∆m1f (0)d (eV) (meV) 2.520 2.499 2.467 2.362 2.326 2.325 2.332 2.342 2.255 2.216 2.112 2.165

23 43 21 34 36 18 10 41 38 1 40 24

a

Theoretical dynamic Stokes shift within the PDRA, calculated according to ∆mPDRA = mPDRA (0) − m1f (also given 1f 1f b in Table 1). First spectral moment of the stationary (relaxed), fluorescence spectrum (at long times). The moments c First spectral moment have been calculated from fits of the monomodal lineshape model to the experimental spectra. of the steady-state, i.e. time-integrated, fluorescence spectrum. The moments have been calculated from the fits of the multimodal lineshape model to the experimental spectra. d ∆m (0) = m (0) − mPDRA (0) gives the discrepancy between the first spectral moment of the time-zero fluorescence 1f 1f 1f of the current approach and that one estimated using the PDRA.

568 569 570

Q(t)

×Er /(Er −

ErCX )

571 572 573 574 575 576 577 578 579 580 581 582

Figure 7: Relaxation functions, Q(t), of coumarin 153 in tetrahy-583 drofuran and dimethylsulfoxide using the model of this work and584 the PDRA. The light red line shows the relaxation function of the585 new approach scaled such, to match the amplitudes of the solvent contribution, i.e. 1 in the PDRA and Er /(Er − ErCX ), for the two586 587 approaches. 588 589 557 558 559 560 561 562 563 564 565 566 567

tion no (dielectric) relaxation (which is assumed to be the590 only one observable in these systems) is possible. This is591 also reflected in the fact of using the emission spectrum592 in a nonpolar solvent as reference for the calculation of593 the time-zero spectrum.[27] In this work, this restriction594 has been lifted and the relaxation is simply connected to595 the frequency spectrum of the system composed of the so-596 lute and the solvent. There are no pure modes related to597 one or the other, but a split between high (mostly inner598 modes) and low (mostly outer modes, though not only) frequencies. 11

4.3.2. The Generalized Langevin Equation Approach The differences with the former model are mostly of energetic nature, i.e. which modes are responsible for the reorganization energy and thus the ensuing free energy surface. A completely different point is touched upon when comparing the present model with a generalized Langevin equation approach (GLEA).[29] In the GLEA the lowfrequency mode relaxation is unimodal but exhibits memory. In principle it seems that both, the GLEA and the model presented here, are able to explain the band position dynamics reasonably well, so the question regarding the most suitable physical picture of the solvent relaxation arises. It would be an interesting exercise to introduce a single mode memory function in the formalism presented here and compare the results with those previously obtained, and vice versa. A clear advantage of the current model over the GLEA is that it can explain the full shape of the bands at different times. The GLEA model as given in Ref. 29 cannot do this as it refers to a given reference lineshape function obtained from the spectrum at very low dielectric constant, very much like the PDRA. On the other hand, the GLEA can be straightforwardly applied to excited state adiabatic transitions, while the present one has not yet been extended to that case. The current approach can, however, very well incorporate underdamped motion over a harmonic free energy surface, meaning a relaxation process with memory. To do this, the relaxation function, Q(t), exploited in the current model should be selected in the appropriate form. Then the present approach, and the one presented in reference 29, should lead to similar results.

599

5. Conclusions

649 650

643

A general analytical expression, eq (1), for the tran-651 sient fluorescence spectrum of a fluorophore in solvents652 is derived, which directly connects an arbitrary solvent653 relaxation function Q(t) with the fluorescence spectrum dynamics. The formation dynamics of a wave packet in the excited state of a fluorophore are calculated using in-654 formation about the pump pulse characteristics.6 As a 655 result, the long known problem[27] of the time zero spec656 trum is circumvented. The presented approach is general657 ized to account for cases where the pumping with excess energy leads to vibrational relaxation in the excited electronic state. In this case, the vibrational relaxation is described in terms of a master equation and fitting to the experimental spectral dynamics potentially provides information not only about the low-frequency mode relaxation function, but also about the relaxation time constants of intramolecular high-frequency vibrational modes. As a matter of fact, we find that it is impossible to separate the low frequency solvent response from the low frequency intramolecular response of the solute probe. Unless intramolecular relaxation was infinitely fast, the dynamics of the latter will always contaminate the dynamics of the former. The fact that the time-zero spectra of the current approach are more blue-shifted than those estimated using the PDRA is a clear indication for this. Probably both, solvent relaxation and intramolecular vibrational redistribution in C153, are too fast for our current set-up. Thus, plenty of the dynamics vanishes within the instrument response function. However, it is clear, that the discrepancy in the position of the time-zero spectra (with respect to the PDRA) becomes greater as the excess excitation energy increases. This points towards the fact that intramolecular low-frequency contributions cannot merely be separated and/or neglected in this type of experiments. The full potential of the here presented approach will be apparent once broadband fluorescence data with significantly improved time-resolution become available, or for systems - unlike C153 - where vibrational redistribution evolves significantly slower. From the theoretical side the question whether vibrationally hot spectra can be simulated, and thus account for the vibrational cooling, may be interesting. Further developments towards a comparison between Markovian multimodal and non-Markovian unimodal solvent relaxation is also desirable.

644

6. Supplementary Material

600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642

645 646 647 648

The Supplementary Material is split into two parts: 1) the theory of time-dependent fluorescence spectra as well as both the experimental and simulated steady-state and fluorescence upconversion spectra of C153 in 13 solvents 6 When the pump pulse can excite a series of the states of the intramolecular high-frequency vibrations, a series of wave packets is created, corresponding to each excited sublevel.

12

as well as the vibrational distributions in the excited state for the wavelength used in the experiment, are presented. 2) An additional file contains the experimental and fitted FLUPS spectra in all solvents at the time-steps chosen for the fitting. Acknowledgements The study was supported by a grant from the Russian Science Foundation (Grant No. 16-13-10122) and the Narodowe Centrum Nauki (SONATA bis No. 2013/10/E/ST4/00534). References [1] G. van der Zwan, J. T. Hynes, Nonequilibrium solvation dynamics in solution reactions, J. Chem. Phys. 78 (6) (1983) 4174– 4185. doi:10.1063/1.445094. [2] H. Sumi, R. A. Marcus, Dynamical effects in electron transfer reactions, J. Chem. Phys. 84 (1986) 4894–4914. doi:10.1063/1.449978. [3] H. Heitele, Dynamic solvent effects on electron-transfer reactions, Angew. Chem. Int. Ed. 32 (3) (1993) 359–377. doi:10.1002/anie.199303591. [4] T. Bultmann, N. P. Ernsting, Competition between geminate recombination and solvation of polar radicals following ultrafast photodissociation of bis(p-aminophenyl) disulfide, J. Phys. Chem. 100 (50) (1996) 19417–19424. doi:10.1021/jp962151n. [5] K. Tominaga, G. C. Walker, T. J. Kang, P. F. Barbara, T. Fonseca, Reaction rates in the phenomenological adiabatic excitedstate electron-transfer theory, J. Phys. Chem. 95 (25) (1991) 10485–10492. doi:10.1021/j100178a040. [6] K. Tominaga, G. C. Walker, W. Jarzeba, P. F. Barbara, Ultrafast charge separation in adma: experiment, simulation, and theoretical issues, J. Phys. Chem. 95 (25) (1991) 10475–10485. doi:10.1021/j100178a039. [7] M. J. van der Meer, H. Zhang, M. Glasbeek, Femtosecond fluorescence upconversion studies of barrierless bond twisting of auramine in solution, J. Chem. Phys. 112 (6) (2000) 2878–2887. doi:10.1063/1.480929. [8] M. Glasbeek, H. Zhang, Femtosecond studies of solvation and intramolecular configurational dynamics of fluorophores in liquid solution, Chem. Rev. 104 (4) (2004) 1929–1954. doi:10.1021/cr0206723. [9] I. A. Heisler, M. Kondo, S. R. Meech, Reactive dynamics in confined liquids: Ultrafast torsional dynamics of auramine o in nanoconfined water in aerosol ot reverse micelles, J. Phys. Chem. B 113 (6) (2009) 1623–1631. doi:10.1021/jp808989f. [10] M. Kondo, I. A. Heisler, J. Conyard, J. P. H. Rivett, S. R. Meech, Reactive dynamics in confined liquids: Interfacial charge effects on ultrafast torsional dynamics in water nanodroplets, J. Phys. Chem. B 113 (6) (2009) 1632–1639. doi:10.1021/jp808991g. [11] Y. Erez, Y.-H. Liu, N. Amdursky, D. Huppert, Modeling the nonradiative decay rate of electronically excited thioflavin t, J. Phys. Chem. A 115 (30) (2011) 8479–8487. doi:10.1021/jp204520r. [12] R. Simkovitch, R. Gepshtein, D. Huppert, Fast photoinduced reactions in the condensed phase are nonexponential, J. Phys. Chem. A 119 (10) (2015) 1797–1812. doi:10.1021/jp508856k. [13] A. V. Barzykin, P. A. Frantsuzov, K. Seki, M. Tachiya, Solvent effects in nonadiabatic electron-transfer reactions: Theoretical aspects, Adv. Chem. Phys. 123 (2002) 511–616. doi:10.1002/0471231509.ch9. [14] S. Thallmair, M. Kowalewski, J. P. P. Zauleck, M. K. Roos, R. de Vivie-Riedle, Quantum dynamics of a photochemical bond cleavage influenced by the solvent environment: A dynamic continuum approach, J. Phys. Chem. Lett. 5 (20) (2014) 3480–3485. doi:10.1021/jz501718t.

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