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Dynamics of Ground State Absorption Spectra in DonorAcceptor Pairs with Ultrafast Charge Recombination Roman G. Fedunov, Anastasiia Victorovna Plotnikova, Vladimir Nikolaevich Ionkin, and Anatoly Ivanovich Ivanov J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b00725 • Publication Date (Web): 16 Feb 2015 Downloaded from http://pubs.acs.org on February 18, 2015

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Dynamics of Ground State Absorption Spectra in Donor-Acceptor Pairs with Ultrafast Charge Recombination Roman G. Fedunov, Anastasiia V. Plotnikova, Vladimir N. Ionkin, and Anatoly I. Ivanov∗ Volgograd State University, University Avenue 100, Volgograd 400062, Russia (Dated: February 16, 2015)

Abstract A theoretical approach to simulation of the transient spectra in molecular systems with ultrafast photoinduced non-radiative electronic transitions is developed. The evolution of the excited and ground state populations as well as the nonradiative transitions between them are calculated in the framework of the stochastic multichannel point-transition model involving the reorganization of the medium and the intramolecular high frequency vibrational modes. Simulations of transient spectra of donor-acceptor pairs excited in the charge-transfer band that are accompanied by ultrafast charge recombination into the ground state demonstrate a possibility of positive band appearance in the transient absorption spectrum caused by those systems in the ground state, which returned there from the excited state. The region of the parameters of the donor-acceptor systems where a positive ground state absorption signal can be observed is discussed. A qualitative comparison of the simulated transient spectra with the experimental data on betaine-30 is presented. Keywords: transient spectrum, charge transfer, intramolecular reorganization, solvent relaxation, stochastic point-transition model, theory of pump-probe experiment



To whom correspondence should be addressed. E-mail: [email protected]

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I.

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INTRODUCTION

Photoexcitation of donor-acceptor pairs in the charge-transfer band in polar solvents, in most cases accompanied by ultrafast charge recombination (CR).1–10 Investigations have uncovered the substantial role of nuclear nonequilibrium in ultrafast CR.9–14 Indeed, in ultrafast CR the nonequilibrium initial nuclear state produced by the pump pulse relaxes in parallel with its chemical transformation, so that the reaction proceeds in a nonequilibrium regime. One of the brightest manifestation of the nonequilibrium regime is a dependence of the effective CR rate constant on the excitation pulse carrier frequency (the spectral effect).10,15–18 Nonexponential kinetics of excited state population decay is another manifestation of the nonequilibrium regime.10,18 Theoretical description of ultrafast chemical transformations is usually aimed at simulation of the population kinetics that is characterized by the only parameter – the rate constant. Even a good fitting of the only parameter does not guarantee that an accepted model adequately describes other characteristics of the chemical transformation. Obviously, much more sophisticated models are required for quantitative simulation of the spectral dynamics obtained in pump-probe experiments. Indeed, the femtosecond spectroscopy provides possibilities of more detailed investigations of ultrafast dynamics of intra- and intermolecular transformations of fluorophores in solutions.19–21 However, the spectroscopic data indirectly characterize the system dynamics and they can be comprehended only in the framework of certain theoretical concepts.22–24 The spectra obtained in pump-probe experiments are composed of several components.19,22,24 Typically, the signals of excited state absorption (ESA), stimulated emission (SE), and bleach (BL) are considered.19 Although the formal theory of pump-probe experiments is well developed,19,22–25 its application to the interpretation of ultrafast photochemical processes encounters grave difficulties caused by several reasons. A pump-probe signal of real molecular systems consists of many spectral bands corresponding to different intramolecular transitions and the number of such bands can be only guessed. Overlap of these spectral bands creates one of them. In particular, relatively weak bands can be completely invisible due to overlapping with stronger bands. To describe the time evolution of the pump-probe signal in systems with ultrafast chemical transformations a detailed theory of such transformations is needed. Typically, an aim of chemical reaction theories is

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the calculation of the rate constant of corresponding reaction21,26–28 while a description of the spectral evolution needs the time-dependent distribution function over the states of the system. This leads to one more difficulty of the theory. A few approaches for description of the influence of the chemical transformation on the transient signals have been suggested. A theory to calculate the pump-probe spectroscopy of the curve-crossing nonadiabatic charge separation processes with harmonic potential surfaces embedded in a dielectric continuum model has been developed to treat charge separation processes in solutions with high friction.29 The theory suggests the simulation of chemical transformations in terms of Zusman’s stochastic approach.30 This approach does not address the reorganization of high-frequency intramolecular vibrational modes that can be of primary importance in ultrafast charge transfer dynamics. A comprehensive model for ultrafast pump-probe spectroscopy of the charge transfer cycle has been suggested in ref 24. This approach provides a complete quantum description of an ultrafast pump-probe charge transfer event in condensed phase. Although, the usage of surrogate Hamiltonian allows consistently treating a number of important aspects affecting the transient spectra, such as initial correlation, non-Markovian dynamics, and explicit description of the pulse field, its application to real systems limited to short-time domain. An influence of strong femtosecond optical pulses on the ultrafast dynamics of molecular systems has been investigated using multilevel Redfield theory applicable for weak system-bath interaction.31 It have been shown that the large amplitude wave-packet motion can significantly modulate the population transfer between the electronic states during the system-field interaction. One more well developed method of pump-probe analysis is ab initio molecular dynamics (MD) ’on the fly’.32,33 The approach is related to the Liouville space theory of nonlinear spectroscopy19 in the formulation given in refs 34–38 involves both simple adiabatic and nonadiabatic dynamics with Tully’s surface hopping algorithms.39 A combination with quantum chemistry methods for electronic structure allows carrying out the multi-state dynamics with precalculated energy surfaces as well as the direct ab initio MD ’on the fly’, in which the forces are calculated, when they are needed in the course of the simulation. In this approach all nuclear degrees of freedom are described in terms of classical mechanics. So the effects of reorganization of intramolecular high-frequency vibrational modes are ignored. The multichannel stochastic model involving the reorganization of the medium and the intramolecular high frequency vibrational modes13,18,40 deals with time-dependent distribu3

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tion functions and, hence, well suited to calculation of the transient signal. This model adequately describes all principal regularities observed in the electron transfer kinetics.41 We mention only the most important: (i) the model predicts the solvent controlled regime in the Marcus normal region and its almost full suppression in the Marcus inverted region as well as a continuous transition between them in the vicinity of the activationless region, (ii) the suppression of dynamic solvent effect is principally caused by the reorganization of high frequency vibrational modes, and (iii) in the inverted region, the multichannel stochastic model predicts the apparent activation energy to be much less than that calculated with Marcus equation42 in full accordance with experimental data43–47 . Apparently, the most direct manifestation of the ultrafast photochemical transformations in pump-probe signal is the hot ground state absorption (GSA). The hot GSA band is caused by the absorption of the systems being in the ground state which come back there from the excited state due to reverse charge transfer. Before thermalization of the particles returned to the electronic ground state this band gives a positive contribution to the pump-probe signal and is placed between BL and SE bands. The last two bands are negative and usually much more intensive, so typically the hot GSA band is not seen as a positive feature in the total signal measured in experiments due to its overlap with BL and SE bands. The aims of this paper are (i) to derive a general expression for the pump-probe signal adapted for the usage of the chemical dynamics simulated in the framework of the multichannel stochastic model, (ii) to investigate the possibility of appearance of the hot GSA band as a positive feature in the pump-probe signal and to clarify the region of the parameters of the donor-acceptor systems where it can be observed, (iii) to qualitatively compare theoretical predictions with the experimental data on betaine-30.48

II.

STOCHASTIC MULTICHANNEL MODEL APPROACH TO SIMULATION

OF TRANSIENT SPECTRA

The photoinduced evolution of a donor-acceptor pair (DAP) is depicted by the following scheme

hν ( + − )FC VR bCT (DA)eq −→ D A e −→ (DA)hot g g

relaxation

solvation

−→

(DA)eq g

(1)

Here photoexcitation of a charge transfer band by a light quantum hν lifts the DAP from the thermalized ground state to the excited Franck-Condon (superscript FC) state. This triggers 4

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several competing processes: solvation, conformational rearrangement of the solute that can involve the solute dipole moment variation, vibrational relaxation (VR) of high frequency intramolecular modes (the term VR includes both stages the intramolecular vibrational redistribution and vibrational cooling) and the back charge transfer (bCT) into the ground electronic state. The bCT produces hot DAPs with nonequilibrium surrounding medium. Finally, these DAPs relax to their equilibrium state. This scheme is close to, but different from that suggested in ref 48 for betaine-30. The evolution of the system on the energetic levels is pictured in Figure 1. We suppose that the solvation of the DAP can be treated in the framework of the Markovian approximation. Relaxation of real solvents are characterized by several relaxation time scales and can be described by the solvent relaxation function, X(t), which is written as a sum of exponentials49–51 X(t) =

N ∑

xi e−t/τi

(2)

i=1

where xi and τi are weight and relaxation time constant of i-th medium mode, respectively. N is the number of the medium (solvent) modes. Conformational rearrangement of the solute can be represented as a strongly damped low-frequency vibrational mode and it can be treated as an additional term in the solvent relaxation function eq 2. The last equation suggests the diffusional motion of each solvent mode, while an initial part of the solvent relaxation is not diffusive but rather inertial.49–51 It was shown that in description of charge transfer kinetics an approximation of the initial relaxation by the exponential function does not lead to an error exceeding the errors caused by the inaccuracy of the measurements of the shortest relaxation time constant.52 Then two parabolic terms for the ground and excited electronic states with the vibrational manifolds can be used to represent the photoinduced evolution of the solute in the framework of linear medium model (Figure 1). The complex motion along the reaction coordinate Q can be replaced by a sum of motions along independent coordinates Qi corresponding to the i-th relaxation mode. The diabatic free energy surfaces for the electronic states in terms of

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the coordinates Qi can be presented in the following form13,40 Ug(⃗n) ⃗ Ue(m)

N M ∑ ∑ Q2i = + nα ~Ωα 4E ri α=1 i=1

=

N ∑ (Qi − 2Eri )2 i=1

4Eri

+

(3)

M ∑

mα ~Ωα + ∆G,

(4)

α=1

where ∆G is the reaction free energy, nα (nα = 0, 1, 2, . . . and α = 0, 1, 2, . . . , M ) are the quantum numbers of the α-th high frequency intramolecular mode with the frequency Ωα , Eri = xi Erm is the reorganization energy of i-th medium relaxation mode, and Erm is the total reorganization energy of the low frequency modes. The vector ⃗n has M components (n1 , n2 , . . . , nα , . . . , nM ). The vector m ⃗ is similar to ⃗n. In the framework of the stochastic point-transition approach18,30,40 the temporal evolution of the system is described by a set of equations for the probability distribution functions for (⃗ n) ⃗ t), and m-th ⃗ sublevel of the excited the ⃗n-th sublevel of the ground electronic state, ρg (Q, (m) ⃗

⃗ t), electronic state, ρe (Q, (⃗ n)

∂ρg ∂t

(m) ⃗ ∂ρe

∂t

ˆ g ρg(⃗n) − = L



(m) ⃗ ⃗ (⃗n) k⃗n m ⃗ (Q)(ρg − ρe ) +

α

m ⃗ ⃗ ˆ e ρ(m) = L − e





⃗ n) ⃗ (m) k⃗n m − ρ(⃗ ⃗ (Q)(ρe g )+

∑ α

⃗ n

1



ρ(⃗nα ) − (nα +1) g



τα

α

1



⃗ α) ρ(m − (mα +1) e

τα

1 (⃗n) ρ , (nα ) g τα

∑ α

1 (m) ρ⃗ , (mα ) e τα

(5) (6)

⃗ stands for the vector with components Q1 , Q2 , ..., QN , L ˆ g and L ˆ e are the Smoluwhere Q chowski operators describing diffusion on the Ug and Ue diabatic terms, correspondingly [ ] N ∑ ⟨ 2⟩ ∂ 2 1 ∂ ˆg = L 1 + Qi + Qi , (7) 2 τ ∂Q ∂Q i i i i=1 [ ] N ∑ ⟨ 2⟩ ∂ 2 1 ∂ ˆe = L 1 + (Qi − 2Eri12 ) + Qi , (8) 2 τ ∂Q ∂Q i i i i=1 where ⟨Q2i ⟩ = 2Eri kB T being the dispersion of the equilibrium distribution along the i-th coordinate. The vector ⃗n′α differs from ⃗n only by the number of vibrational quanta for αth mode ⃗n′α = (n1 , n2 , . . . , nα + 1, . . . , nM ). So, the model accounts for the reorganization of a number of intramolecular high frequency vibrational modes that generally leads to the vibrational sublevels of both the excited and ground states. The vibrational sublevels of the excited state can manifest themselves on very short time scale due to their fast relaxation. Figure 1 shows a number of vibrationally excited sublevels of the excited and ground electronic states. 6

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Direct and back electron transitions between the vibrational sublevels of the excited state and a vibrational sublevel of the ground state are described by the Zusman parameters ( ) 2πV⃗n2m 2πV⃗n2m ⃗ 2 ⃗ ⃗ δ Ug(⃗n) − Ue(m) = δ (Y − Y⃗nm V⃗n2m (9) ⃗ ), nm ⃗, ⃗ = V F⃗ ~ ~  2 √ nα +mα −2r min(nα ,mα ) nα −r ∑ ∏ (−1) ( Sα )  , Sα = Ervα , = exp {−Sα } nα !mα !  r!(nα − r)!(mα − r)! ~Ωα α r=0

k⃗nm ⃗ = F⃗nm ⃗

∑ where V is the electronic coupling, Y = i Qi is the collective energetic reaction coordinate, ∑ (⃗ n) (m) ⃗ Y⃗nm and Ue . ⃗ = Erm + ∆G + α (nα − mα )~Ωα is a point of the intersection of terms Ug F⃗nm ⃗ is the product of the Franck-Condon factors, Sα = Ervα /~Ωα and Ervα are the HuangRhys factor and the reorganization energy of the αth high frequency vibrational mode, ∑ respectively. Erv = α Ervα is the total reorganization energy of the high frequency vibrational modes. A single-quantum mechanism of high frequency mode relaxation is adopted (nα )

and the transitions nα → nα − 1 are supposed to proceed with the rate constant 1/τα (nα )

where τα

(1)

= τα /nα .

To specify the initial conditions, we assume that the pump pulse has a Gaussian form { } t2 E(t) = E0 exp iωe t − 2 (10) τe and its duration is short enough so that the medium is considered to be frozen during excitation. All high frequency vibrational modes are supposed initially to be in the ground state. This allows us to obtain the following general expression for the initial probability distribution function on the excited term16 } { ∑e 2 2 ∑ (m) ⃗ e2 Q ] τ Q − [~δω e i e i ⃗ ⃗ t = 0) = APm − ρ(em) (Q, ⃗ exp − 2~2 4Eri kB T  [ ]2  [N ] (m) ⃗    ~δωe  ∏ S mα e−Sα α = Pm − exp ⃗   mα ! 2σ 2   α=1

(11)

(12)

N ∑ ⃗ ei = Qi − 2Eri , ~δωe(m) where Q = Erm + ∆G + mα ~Ωα − ~ωe , σ 2 = (2Erm kB T ) + ~2 τp−2 , α=1

and Pm ⃗ is a factor proportional to the fraction of the excited DAPs in the vibrational state with vibrational quantum numbers stated by the vector m. ⃗ The factor A depends on the power and duration of the pump pulse and determined by eq 13 ∫ ∏ ∑ ⃗ ⃗ t = 0) (Q, dQi Wp = ρ(em) (m1 ,m2 ,...,mN )

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(13)

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where Wp is the probability of the electronic excitation of DAP by the pump pulse. The (⃗ n) ⃗ initial condition for the ground state distribution function, ρg (Q, t = 0), can be written as the difference: n) ⃗ ρ(⃗ g (Q, t

[ ] (eq) ⃗ 0 ⃗ = 0) = δ⃗n,⃗0 ρg (Q) − ρe (Q, t = 0)

(14)

where δ⃗n,⃗0 is the Kronecker symbol and ⃗ ρ(eq) g (Q)

=

∏ i

{ } ∑ 1 Q2i ⃗ t = 0) = √ exp − , ρ0e (Q, 4Eri kB T 2πEri kB T (m ,m ,...,m 1

2

⃗ ⃗ t = 0) ρ(em) (Q, N)

(15) are the equilibrium distribution function in the ground state and the distribution of population transferred to the excited state, correspondingly. The system of differential equations (5)-(6) with the initial conditions (11) and (14) was solved numerically using the Brownian simulation method.16,53 To calculate the kinetics of electronic state populations with satisfactory accuracy 105 trajectories are needed. To simulate transient spectra we use from 106 to 107 trajectories because the spectral range is divided into 400 intervals. This provides an acceptable accuracy except extremely small signals. The transient absorption signal in the framework of the considered model is determined by the following expression ∆A = −ASE + AGSA − ABL , where Ab (ωp , τ ) = C −1 ωp

∑∑ ⃗ n

m ⃗

∫∞ F⃗n,m ⃗ −∞

(16)

 [ ]2  (m) ⃗ (⃗ n)   Ug − Ue + ~ωp  ∏ ⃗ τ ) exp − dQi (17) ρb (Q,   2~2 τp−2  i 

⃗ ⃗ τ ) = ρ(em) ⃗ τ ), ρBL (Q, ⃗ τ) = The index b runs through {SE, GSA, BL}, where ρSE (Q, (Q, (eq) ⃗ (⃗ n) ⃗ ρg (Q) − ρg (Q, τ ) is the distribution of the holes on the ground state term produced ⃗ τ ) is the distribution of the particles on the ground state by the pump pulse, and ρGSA (Q,

term at time τ transferred from the excited state (see Figure 2). The transient absorption spectrum, Ab , is represented by the overlap integral of the probability distribution function in the corresponding term and the probe pulse applied with the time delay τ . Here ωp and τp are the carrier frequency and duration of the probe pulse. 8

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III.

TRANSIENT SPECTRA OF DONOR-ACCEPTOR PAIRS WITH ULTRA-

FAST BACK ELECTRON TRANSFER

In this section a manifestation of hot GSA band in transient absorption spectra for molecular systems with ultrafast photoinduced electronic transitions is investigated. To connect the simulations with real systems, we reference to a well studied intramolecular DAP, namely the betaine-30 in acetonitrile (ACN). The parameters of this system reported in ref 48 are: ∆G = −1.35 eV, Erm = 0.45 eV, x1 = 0.34, τ1 = 0.065 ps, x2 = 0.16, τ2 = 0.6 ps, x3 = 0.5, τ3 = 10.0 ps, Erv = 0.18 eV, Ω = 0.17 eV, τv = 0.01 ps, ωe = 1.95 eV, τe = 0.06 ps, τp = 0.06 ps, kB T = 0.025 eV. Here the low frequency (conformational) intramolecular contribution is considered as an addition to the medium reorganization energy with the wight x3 and the relaxation time constant τ3 . This mode is connected with geometrical relaxation within the S1 state of betaine-30 from a planar toward an orthogonal arrangement of the phenolate and the pyridinium ring.54 Since there is no direct information on the value of the electronic coupling, V , it is considered to be a free parameter. The transient absorption spectra calculated with these parameters are depicted in Figure 3. It is suggested that the excited state absorption band is not overlapped with the hot GSA and is not considered for simplicity of the analysis. Frames A and B show the total transient spectra and their constituents (SE, BL, and GSA) at the early times from 0.05 to 0.70 ps. The total signal is characterized by two negative bands. The BL is seen in the region around 630 nm. Its minimum shifts to the blue with time. Broad band of SE is placed in the red with the minimum position moving from 800 to 1000 nm. The reasons of these changes are clarified by the spectral dynamics of the constituents shown in Frame B. In particular, the BL band changes with time are minor and the BL shift in the total spectrum is caused by the increase of the hot GSA band intensity. The time-dependent shift of the SE band mainly reflects the relaxation of the Franck-Condon excited state. The trends obtained in simulations are in qualitative agreement with experimental results but there are differences. In particular, in the total spectrum the SE minimum position shifts to the red (by 1000 cm−1 on the times from 50 to 100 fs) in accordance with the experiment (see Figure 3, frame b in ref 48). Simultaneously with the shift, the BL and SE band intensities reduce by one third. This is slower than the experimental BL decay. Another discrepancy is that in the experiment the SE band is narrower (placed from 800 to 950 nm) 9

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and in the region from 700 to 800 nm a positive band dominates at the times from 0.2 to 1.0 ps. The calculations show (frame B) that in this time interval hot GSA is rather weak and cannot lead to a positive band in the total signal. It was supposed that this positive band in betaine-30 is caused by absorption of a dark state48 which is not considered in this paper. At larger times from 0.7 to 4.0 ps the BL and SE intensities (frame C) demonstrate dramatic decay that is as strong as that in the experiment.48 In simulated transient spectra a positive peak appears in the area of 750 nm later 1 ps and its intensity is small in contrast with BL. This finding is in accord with the experimental data on betaine-30 that also show an analogous positive band (see Figure 2, frame d in ref 48). The positive and negative band intensities are equalized at the times t > 2 ps. Frame D shows the SE intensity decay and the hot GSA intensity growth that reflect the progress of the back electron transfer. Simultaneously hot GSA maximum moves to the blue from 715 to 655 nm. The BL minimum is invariable around 645 nm. However in Frame C the BL minimum in the total spectrum moves to the blue from 630 to 590 nm due to a variation of the GSA constituent. The integral intensity of the total signal decreases to zero when both hot GSA and BL peaks in frame D approach to 645 nm. It means that the back electron transfer has been terminated and the hot ground state has been thermalized. Both the ultrafast back electron transfer and the slow-relaxing mode leading to slow equilibration of the hot ground state population play a key role in the formation of the positive band in Frame C. A retardation of the hot ground state thermalization relative to the fast back electron transfer occurs due to the slow relaxation of the mode with timescale, τ3 = 10 ps. It is confirmed in frame D, where hot GSA and BL integral intensities are aligned earlier than their peak positions. The timescale of the back electron transfer estimated from the experimental data,48 τbET = 1.2 ps, is obtained in the simulations at the extremely large value of the electronic coupling, V = 0.8 eV, that is far beyond the applicability of the present theory. Parameters mentioned earlier and considerably smaller and, hence, more realistic electronic coupling would lead to much slower electron transfer and the positive band would disappear in the total signal at all. A way out can be found if one suppose that the reorganization energy of the high frequency mode48 has been underestimated. Really, the estimation of the medium reorganization energy of betaine-30 in acetonitrile varies from 800 to 6000 cm−1 .12,55–59 More10

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over the TA spectra recorded with pyrylium phenolate in ACN60 are similar to those recorded with betaine-30 in ACN, the main difference being the pronounced vibrational structure of the bleach. This means that in this system the total reorganization energy mainly consists of the reorganization energy of the high frequency vibrational modes. The total reorganization energy accompanying an electronic transition is reliably determined from optical spectra and only its division into high and low frequency parts may be questionable. Accounting for this reason we increase Erv and use alternative parameters: ∆G = −1.35 eV, Erm = 0.23 eV, x1 = 0.34, τ1 = 0.065 ps, x2 = 0.16, τ2 = 0.6 ps, x3 = 0.5, τ3 = 5.0 ps, Erv = 0.40 eV, Ω = 0.17 eV, τv = 0.5 ps, ωe = 1.95 eV, τe = τp = 0.06 ps, kB T = 0.025 eV. For these parameters the timescale of the back electron transfer τbET = 1.0 ps is obtained with considerably smaller electronic coupling V = 0.10 eV that is in the area of the applicability of the theory. This result is expected because the high frequency modes are well known to accept the electronic energy much more effective than the low frequency modes. With these parameters vibrational structures are seen in the spectra up to 4 ps (see Figure 4). At early times this structure is similar to that observed in the experiments.48 To appreciate the origin of this vibrational structure the constituents (SE, BL, and GSA) of the total spectrum are shown in Figure 4 (frames B, D, E, and F). One can see that the vibrational structure of BL is short-living and disappears at t = 0.3 ps while in SE and hot GSA it is kept up to 4 ps. Long-lived vibrational structure in hot GSA is observed only in the long-wavelength area where the back charge transfer is still in progress. The reason of this difference is as follows: the BL band structure blurring is determined by thermalization of the holes in the ground state and it occurs at the stage of the relaxation of the fastest medium mode with the time constant, τ1 . The structure of the SE and hot GSA signals are maintained by back charge transfer proceeding through the sinks placed at the points Y = Y⃗nm ⃗ separated by the distance ~Ω. The periodic modulation of the populations of the excited and ground states manifests itself in modulation of the transient SE and hot GSA signals. It should be noted that small-grained structures seen in frame C for t = 3 and 4 ps has statistical origin and caused by insufficient number of the survived trajectories.

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IV.

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CONDITIONS OF POSITIVE HOT GSA BAND OBSERVATION

To quantify the conditions of appearance of a positive hot GSA band in transient spectra in Figure 5 the dependencies of the maximum magnitude of hot GSA band (hm ) at time delay t = 4 ps on the electronic coupling, V , are pictured for four values of the relaxation time constant of the conformational intramolecular mode, τ3 . Each curve has a maximum and with increasing τ3 its position shifts to the area of larger V . When V value is small the recombination proceeds in a vicinity of the excited state free energy minimum due to smallness of the back electron transfer rate compared with the relaxation rate 1/τ3 . At the time interval t < 4 ps the ground state population recovery is small and hot GSA signal is hidden by grater negative BL and SE signals. Increasing hm with increase of the V magnitude is a direct consequence of increasing the recombination rate constant with increase of the electronic coupling that leads to growth of the population of the ground state in the area between the minima of the Ug and Ue terms (see Figure 1). With further increase of V the hot transition efficiency to the high excited vibrational sublevel of the ground state increases14 and a wave packet is formed closer to the ground state term minimum. This results in larger overlapping of the hot GSA and BL signals and suppression of hot GSA in the total spectrum. The time point of the positive hot GSA signal appearance depends on the model parameters and can be considerably different from t = 4 ps accepted in Figure 5. Since the electronic coupling V is determined with a huge uncertainty it is interesting to obtain a connection between the magnitude of V and the time point of positive hot GSA signal appearance. In Figure 6 the hot GSA maximum magnitude as a function of the electronic coupling and the time delay, t, is pictured. The simulations show the presence of absolute maxima at the times that well correlate with relaxation time of intramolecular mode, τ3 . This result is expected because the weight of the third mode x3 is large and its relaxation time τ3 essentially limits the duration of the thermalization of the hot ground state. On the other hand, increasing τ3 leads to decreasing V values at which the positive hot GSA maximum is achieved. This reflects an obvious trend of decreasing the required rate constant of the back electron transfer with increasing the nuclear relaxation time. To demonstrate that a positive hot GSA maximum can be observed for different relations between the reorganization energies of the quantum and classical modes in Table the values

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of the absolute hot GSA maximum, hm , are presented. This Table shows that both hm and kbET monotonically raise with decreasing τ3 . When the magnitudes of Erm and Erv varies, the V value corresponding to the absolute hot GSA maximum adjusts so that the kbET is kept practically invariable. The variation of the time point of the absolute hot GSA maximum achievement, τm , closely follows to τ3 alteration. Results presented in Table reveal a rather wide area of model parameter values in which a positive hot GSA maximum can be observed. However the key parameters are τm and closely related with it the rate constant kbET . Positive hot GSA signal appears on isolines that are similar to hyperboles. Indeed, the relation of V and the reverse of the time delay along these lines are straight lines presented in Figure 7. The reason of linear correlation can be comprehended if one takes into account that the dependence of kbET on the V value is close to linear in the range of interest (see Figure 8). Naturally, in the non-adiabatic limit the dependence of kbET on the V is quadratic. However in the region of ultrafast back electron transfer interplay between dynamic solvent effect and a number of reaction sinks leads to nearly linear dependence. So, almost straight lines in Figure 7 reflect a linear correlation between kbET and the time point of the positive hot GSA signal appearance. Main conclusion ensuing from the results presented in this subsection is: for x3 ≥ 0.5 (i) the positive hot GSA maximum is observable for τ3 kbET & 1; (ii) the absolute hot GSA maximum is achieved at τm ≃ τ3 .

V.

CONCLUDING REMARKS

A theory of time-dependent absorption spectra of donor-acceptor pairs with ultrafast photochemical transformations is developed. The multichannel stochastic model involving the reorganization of the medium and the intramolecular high frequency vibrational modes is used for calculations of the time-dependent distribution functions. The calculations of transient spectra of the donor-acceptor pairs excited in the charge-transfer band and accompanied by ultrafast charge recombination into the ground state of the pair are shown that the appearance of a positive band in the transient absorption spectrum caused by those systems in the ground state, which returned there from the excited state. This band is located between the bands of the bleach and the stimulated emission. The results of calculations 13

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qualitatively agree with the experimental data for the transient spectra of betaine-30. In this paper we did not try to quantitatively fit the simulations to the betaine-30 transient spectra because there is a number of overlapping bands. Among them ESA and so called dark state absorption (DSA) of unknown nature. ESA in betaine-30 should not strongly affect the hot GSA signal although it can noticeably suppress the BL. On the other hand, in early times there is a strong overlap of the DSA and hot GSA bands that complicates the interpretation of the spectral dynamics. A simulation of ESA and DSA can be fulfilled, but it requires too many fitting parameters. One of the most important uncertainty is connected with the angle between reaction coordinates corresponding to different electronic transitions at the pump and probe stages.61 An experimental observation of a positive hot GSA band in transient spectra has one more important aspect. Such observation could clarify the mechanism of ultrafast charge recombination in excited donor-acceptor complexes which demonstrate unexpected dependence of the CR rate constant on the reaction free energy:1,2 the logarithm of the charge recombination rate constant decreases monotonously, nearly linearly, with increasing the reaction exergonicity whereas the standard equilibrium Marcus nonadiabatic theory predicts a bell-shaped dependence.42 The linear dependence observed in the experiments can by well reproduced in the framework of the multichannel stochastic model under supposition that the vast majority of the initially excited donor-acceptor complexes recombine in the hot regime when the nonequilibrium wave packet passes through a number of the term crossings corresponding to transitions toward vibrational excited states of the electronic ground state.62 This mechanism is very similar to the mechanism of hot GSA positive band formation in the transient absorption spectrum. Indeed, hot GSA as a positive band in the transient absorption spectrum of donor-acceptor complexes can appear if the back electron transfer is ultrafast and its time constant is shorter than the medium relaxation time. In this case the back electron transfer should mainly proceed in the hot regime. Acknowledgment. This work was supported by the Russian foundation for basic research (Projects No 13-03-97062 and No 14-03-00261).

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References and Notes

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Letrun, R.; Koch, M.; Dekhtyar, M. L.; Kurdyukov, V. V.; Tolmachev, A. I.; Rettig, W.; Vauthey, E. Ultrafast Excited-State Dynamics of Donor-Acceptor Biaryls: Comparison between Pyridinium and Pyrylium Phenolates. J. Phys. Chem. A 2013, 117, 13112–13126.

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TABLE I: The time point of absolute hot GSA maximum achievement and the back electron transfer rate constant for a number of model parameters (Erm , Erv , τ3 )

No

Er

τ3 , ps V , eV τm , ps kbET , ps−1 hm 1.0

0.085

1.3

1.27

0.137

1 Erm =0.23 2.0

0.08

2.2

1.0

0.120

0.035

7.6

0.29

0.104

10.0 0.035

10.0

0.24

0.090

1.0

0.19

1.3

1.36

0.309

2 Erm =0.34 2.0

0.115

2.5

0.69

0.277

0.05

7.6

0.21

0.237

10.0 0.035

15.7

0.11

0.230

1.0

0.415

1.3

1.06

0.526

3 Erm =0.45 2.0

0.31

2.2

0.89

0.492

5.0

0.105

7.9

0.18

0.426

10.0

0.09

14.2

0.12

0.421

Erv =0.40

Erv =0.29

Erv =0.18

5.0

5.0

Figure captions Figure 1. Schematic representation of free energy curves. The dashed lines are the vibrational sublevels of the ground and excited electronic states. The arrows stand for directions of the medium relaxation and the wavy arrows for the vibrational relaxation. The population densities on some sublevels are shown as gray packets. Figure 2. Schematic representation of wave packet dynamics on free energy curves and the transitions observed in the transient absorption spectra. The particle and hole population 21

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densities are pictured with filled and empty wave packets, correspondingly. The arrows stand for directions of the transitions to different vibrational sublevels. The signs of the constituents in the total transient absorption spectra are indicated in the circles. Figure 3. The transient absorption spectra (∆A(λ) = ∆A(ωp ) and λ = 2πc/ωp ) calculated with the parameters of the betaine-30 in ACN (frames A and C), their constituents (frames B and D), and the experimental data borrowed from ref 48 (frames F and E). The parameters are listed in the text. Figure 4. The transient absorption spectra, ∆A(λ). The parameters are the same as in Figure 3 except for V = 0.023 eV, Erm = 0.23 eV, and Erv = 0.40 eV. Figure 5. Hot GSA maximum magnitude, hm , as a function of electronic coupling, V . The parameters are the same as in Figure 4 except for τ3 and V . The values of τ3 are: (1) – τ3 = 1 ps, (2) – τ3 = 2 ps, (3) – τ3 = 5 ps, (4) – τ3 = 10 ps. Figure 6. Hot GSA maximum magnitude as a function of electronic coupling, V , and the time delay, t. The parameters are the same as in Figure 4 except for τ3 . The values of τ3 are: (1) – τ3 = 1 ps, (2) – τ3 = 2 ps, (3) – τ3 = 5 ps, (4) – τ3 = 10 ps. Figure 7. Reciprocal time delay, t0 , at which the hot GSA signal becomes positive as a function of the electronic coupling, V . The parameters are the same as in Figure 4 except for τ3 and V . The values of τ3 are: (1) – τ3 = 1 ps, (2) – τ3 = 2 ps, (3) – τ3 = 5 ps, (4) – τ3 = 10 ps. Figure 8. Hot GSA maximum magnitude, hm , as a function of the electronic coupling, V . The parameters are the same as in Figure 4 except for τ3 and V . The values of τ3 are: (1) – τ3 = 1 ps, (2) – τ3 = 2 ps, (3) – τ3 = 5 ps, (4) – τ3 = 10 ps.

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(m)

U

e

+

-

FC

(D A )

e

(n)

U

g

h

U

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hot

(DA)

g

eq

(DA)

g

reaction coordinate

FIG. 1: Fedunov, Plotnikova, Ionkin, Ivanov

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(m)

U

e

-

+

SE

g

BL

(n)

U

hot GSA

U

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

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-

reaction coordinate

FIG. 2: Fedunov, Plotnikova, Ionkin, Ivanov

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0.2

0.7

A

hot GSA

B

hot GSA

0.5

0.0

E

0.4

0.4 0.7

0.3

-0.2

A

0.5

0.3

0.5 0.3 0.2 0.1

0.05

-0.8

-1.0

0.2

0.7

-0.4

0.1

-0.8

0.5

0.0

SE

0.3 0.2

-0.6

0.2

0.2 0.1

0.0

0.7

-0.4

-0.2

SE

0.1

0.05

-0.4

BL

0.05

-1.2

BL

-1.2

-0.6

-1.4 400

800

1200

1600

2000

400

0.2 C

hot GSA

800

1200

1600

0.0

3.0 2.0

1.0 BL

F

0.7

0.2

0.7

1.0 2.0

0.1

3.0

2.0 0.7

-0.4

4.0

0.0

SE

-0.1

-0.8

-0.6

900

0.3

1.0

-0.4

800

D

0.0

SE

700

1.0

0.4

-0.2

600

hot GSA

0.8

4.0

500 0.4

4.0

1.2

2000

2.0

A

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

The Journal of Physical Chemistry

BL

-0.2

-1.2

0.7

-0.8 400

800

1200

1600

2000

400

800

1200

1600

Wavelength / nm

Wavelength / nm

FIG. 3: Fedunov, Plotnikova, Ionkin, Ivanov

25

ACS Paragon Plus Environment

2000

500

600

700

800

Wavelength / nm

900

The Journal of Physical Chemistry

E

A

B

0.7

0.0

0.4

0.5

hot GSA

0.2

0.0

0.5

-0.4

0.0

0.3

0.7

-0.2

0.05

0.3

A

0.2

0.7

SE

0.1

-0.4 0.5

-0.4

0.05

-0.8

0.3

-0.8 BL

-1.2

SE

0.2

-0.6

0.1

-1.2 BL

0.05

-0.8 400

800

1200

1600

hot GSA

2000

C 4.0

0.0

400

800

1200

800

3.0

2.0

0.8

1200

1600

D

4.0

1.2

1600

2000 F

4.0

0.0

2.0

hot GSA

3.0

1.0 2.0

-0.1

0.7

0.4

-0.2

A

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

Page 26 of 31

1.0 SE

0.0

0.7

-0.4

-0.2 -0.4

1.0 SE

-0.3

-0.8

-0.6 BL

BL

0.7

-1.2 -0.4

400

800

1200

1600

2000

400

800

1200

1600

Wavelength / nm

Wavelength / nm

FIG. 4: Fedunov, Plotnikova, Ionkin, Ivanov

26

ACS Paragon Plus Environment

800

1200

1600

Wavelength / nm

2000

Page 27 of 31

0.15

0.10

hm

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

The Journal of Physical Chemistry

2

1

0.05

3

0.00 4

-0.05

0.05

0.10

V, eV

FIG. 5: Fedunov, Plotnik0va, Ionkin, Ivanov

27

ACS Paragon Plus Environment

-0.4270 -0.3559

=1 ps

0.1

-0.2848

3

=2 ps

0.1

3

-0.07137 0.075

-0.1650

0.075

-0.03581

-0.09325

-2.500E-4 0.07088

-0.02150 0.01438

0.05

0.05

0.05025

0.08866 0.1064 0.1242

0.08613 0.1041

0.025

0.025

0.1130

0.1331

0.1220

0.1420

-0.4660 -0.3945

1.3

2.8

4.3

5.8

7.3

=5 ps

0.1

1.3

8.8

3

2.8

4.3

5.8

7.3

8.8

=10 ps

0.1

3

-0.3302

-0.2515

-0.2599 0.075

0.075

-0.1895 -0.1191

-0.1085

-0.04875

-0.03700 0.03450 0.07025

0.05

-0.01356

0.05

0.02163 0.05681

0.08813 0.09706 0.1060

-0.4710 -0.4006

-0.3230 -0.1800

-0.3803 -0.2367

-0.1069

0.03531

-0.4520 -0.3085

-0.1958

V, eV

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

Page 28 of 31

V, eV

The Journal of Physical Chemistry

0.07441 0.025

0.08320

0.025

1.3

2.8

4.3

5.8

time, ps

7.3

8.8

0.09200

1.3

2.8

4.3

5.8 time, ps

FIG. 6: Fedunov, Plotnikova, Ionkin, Ivanov

28

ACS Paragon Plus Environment

7.3

8.8

Page 29 of 31

1 1.5 -1

2

0

1/t , ps

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

The Journal of Physical Chemistry

1.0

3 4

0.5

0.00

0.02

0.04

0.06

0.08

0.10

V, eV

FIG. 7: Fedunov, Plotnikova, Ionkin, Ivanov

29

ACS Paragon Plus Environment

, ps

Page 30 of 31

1.5

3

2

1

bET

k

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

-1

The Journal of Physical Chemistry

4

1.0

0.5

0.0 0.02

0.04

0.06

0.08

0.10

0.12

V, eV

FIG. 8: Fedunov, Plotnikova, Ionkin, Ivanov

30

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Page 31 of 31

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

The Journal of Physical Chemistry

hot GSA hot GSA

SE

SE

BL

BL

400

800

1200

1600

400

800

1200

Wavelength / nm

FIG. 9: TOC

31

ACS Paragon Plus Environment

1600

Dynamics of ground state absorption spectra in donor-acceptor pairs with ultrafast charge recombination.

A theoretical approach to simulation of the transient spectra in molecular systems with ultrafast photoinduced nonradiative electronic transitions is ...
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