Dynamic Solvent Effect on Ultrafast Charge Recombination Kinetics in

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Dynamic Solvent Effect on Ultrafast Charge Recombination Kinetics in Excited Donor-Acceptor Complexes Tatyana Vladimirovna Mikhailova, Valentina A. Mikhailova, and Anatoly Ivanovich Ivanov J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b09363 • Publication Date (Web): 03 Nov 2016 Downloaded from http://pubs.acs.org on November 3, 2016

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The Journal of Physical Chemistry

Dynamic Solvent Effect on Ultrafast Charge Recombination Kinetics in Excited Donor-Acceptor Complexes Tatyana V. Mikhailova, Valentina A. Mikhailova, and Anatoly I. Ivanov∗ Volgograd State University, University Avenue 100, Volgograd 400062, Russia E-mail: [email protected]

Abstract Manifestation of the dynamic solvent effect on charge recombination kinetics of photoexcited donor-acceptor complexes in polar solvents has been investigated within the framework of the multichannel stochastic model. The model takes into account the reorganization of both the solvent and a number of the intramolecular high-frequency vibration modes as well as their relaxation. The non-Markovian solvent dynamics are described in the terms of two relaxation modes. The similarities and differences inherent to ultrafast charge transfer reactions occurring in the nonequilibrium and thermal regimes have been identified. The most important differences are: (i) the dynamic solvent effect is strong in the area of weak exergonicity and is weak in the area of strong exergonicity for thermal reactions while for the nonequilibrium reactions the regions of strong and weak dynamic solvent effect are reversed; (ii) increasing the electronic coupling value results in a decrease of the dynamic solvent effect magnitude for nonequilibrium electron transfer and in its increase for the thermal reactions; (iii) two-staged regime most clearly manifests if the reorganization energy of the relaxation modes noticeably ∗ To

whom correspondence should be addressed

1 ACS Paragon Plus Environment


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exceeds the charge recombination free energy gap. With the rise of the electronic coupling the kinetics approach to the exponential regime since in the limit of strong electronic coupling the reaction includes only single, nonequilibrium, stage.

Introduction The theoretical investigations of electron transfer (ET) reactions in polar solutions started with Marcus’s theory 1 that was generalized in the early 1980s to account for dynamic solvent effect (DSE) on ET kinetics. 2–4 This triggered extensive experimental and theoretical studies of charge transfer in solutions. 5–19 Theory of DSE 2–4,18,20 allowed bridging between nonadiabatic and solvent controlled regimes and answering the question when the solvent relaxation, rather than microscopic electronic processes, constitutes the rate-determining step of ET. 3 In the framework of the stochastic pointtransition model the ET rate constant is described by the equation 3 Vel2 kET = h¯ (1 + g)

√

] [ π (∆G + Erm )2 exp − Erm kB T 4Erm kB T

(1)

where Vel is the electronic coupling, h¯ is the Planck constant, kB is the Boltzmann constant, T is the temperature, Erm is the solvent reorganization energy, [ ] 2π Vel2 τL 1 1 g= + h¯ |∆G + Erm | |∆G − Erm |

(2)

is the Zusman parameter determining crossover from the Golden Rule to the solvent control, ∆G is the reaction free energy change, τL is the longitudinal dielectric relaxation time. From Eq. (1) several important conclusions follow: (i) in the nonadiabatic limit, g ≪ 1, the ET rate constant is proportional Vel2 and is independent of the longitudinal relaxation time of the solvent, τL ; (ii) in the opposite, solvent controlled, limit, g ≫ 1, the ET rate constant is inversely proportional to the longitudinal relaxation time of the solvent and is independent of the electronic


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coupling, that is, the dependence kET (Vel ) is saturated; (iii) the upper limit for the ET rate is achieved in the activationless region where it is close to 1/τL for polar solvents with Erm ≈ 1 eV. These predictions were partly confirmed in a series of experimental studies. 21–29 Later, it was shown that the behavior of real systems can considerably deviate from these predictions. In particular, the observed rate constant may exceed the upper limit, 1/τL , by several orders of magnitude. 27,30–37 Besides, it is appeared that the DSE is observed for relatively slow reactions occurring in the Marcus normal region and it is absent for fast ET occurring in the activationless or inverted regions. 19 All the regularities were shown to emerge from the multichannel stochastic point-transition model. 38,39 It should be stressed that these conclusions were obtained for charge transfer occurring in the equilibrium regime when the initial state of the solvent and intramolecular degrees of freedom are close to the thermal equilibrium. At the same time there is a wide class of important processes in which nonequilibrium charge transfer plays a central role. Such a nonequilibrium of the nuclear subsystem can be created by both the photoexcitation and chemical transformations. The most studied nonequilibrium ET reaction is charge recombination (CR) in photoexcited donor-acceptor complexes (DAC). 40–44 Indeed, photoexcitation of DACs by a short laser pulse in a charge transfer band triggers a series of processes in accord with scheme hν

IP D+ + A− DA ↔ D+ A− −→

k

kCR

(3)

where hν indicates the photoexcitation of the ground neutral state of a complex, DA, leading to the population of an excited state that is the charge separated state of DAC, D+ A− , kCR is the CR rate constant, and kIP is the rate constant of the free radical-ion generation. The nonequilibrium CR has several peculiarities that considerably differ from that observed in thermal ET reactions. Among CR kinetics peculiarities the most famous is the lack of the Marcus normal region in the free energy gap law 41–44 whereas the standard equilibrium Marcus nonadiabatic theory predicts a bell-shaped dependence. 1 An explanation of the monotonic decrease of



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the CR rate constant with free energy gap rise based on the fact that the laser pulse populates a nonequilibrium initial vibrational state of the DAC has been proposed in ref 45. Further development of this idea showed that the reorganization of the high-frequency vibrational modes plays a paramount role strongly increasing the rate constant of not only strongly but also weakly exergonic nonequilibrium CR. 46–50 Hence, for quantitative description of the CR kinetics of the photoexcited DACs their real spectrum of intramolecular high-frequency modes should be employed. Such an intracomplex vibrational spectrum typically includes 5 – 10 active modes. 49,51 The decay of excited vibrational states due to intramolecular vibrational redistribution and relaxation can also significantly affect the CR kinetics. 48,50 The stochastic model involving reorganization of the solvent and several intramolecular high-frequency vibrational modes of a DAC was able to fit the CR rate constant dependence on the free energy gap 52,53 obtained in the experiments. 41–44 The nonequilibrium ultrafast CR is a key elementary stage of diverse photoinduced ET. Often the CR can lead to the loss of the energy or selectivity of a photoreaction. In particular, the ultrafast CR considerably limits the efficiency of photovoltaic devices based on the dye-sensitized solar cells 54 and photocatalysis. 55 In such processes the CR is undesirable process and it is very important to know how to suppress it. For efficient control of the photoreaction effectiveness a knowledge of detailed CR mechanism is needed. The aim of this paper is to investigate the regularities of the dynamic solvent effect on the reactions occurring in a nonequilibrium mode. As a specific process the CR in the photoexcited DACs is considered. For modelling the nonequilibrium CR kinetics a stochastic model involving the reorganization of the solvent with two relaxation time scales and the reorganization of several intramolecular high-frequency vibrational modes is used.

Theory and Computational Details Within framework of the stochastic point-transition approach 3,4,38,39,46,47,49,52,56–58 CR kinetics of the excited DACs can be described by a set of differential equations for the probability distribution


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functions ( ) ∂ ρR (⃗n) = Lˆ R ρR − ∑ k⃗n (Q1 , Q2 ) ρR − ρP ∂t n1 ,n2 ,...,nM (⃗n)

∂ ρP ∂t

( ) (⃗n) (⃗n) = Lˆ P ρP − k⃗n (Q1 , Q2 ) ρP − ρR + ∑

(4) 1

(nα α τvα

(⃗n′α )

ρ +1) P

−∑

1

(⃗n) ρ (nα ) P α τvα

(5)

(⃗n)

Here ρR (Q1 , Q2 ,t) and ρP (Q1 , Q2 ,t) are the probability distribution functions for the electronic excited state (reactant) formed by a short laser pulse in the charge transfer band and the neutral ground state (product) with the excitation of nα (nα = 0,1,2,...) vibrational quanta for α th intramolecular DAC modes with the frequency Ωα (α = 0, 1, 2, ..., M). The vector ⃗n has Mcomponents (n1 , n2 , ..., nα , ..., nM ). The vector⃗n′α differs from⃗nα only by the number of vibrational quanta for α th mode⃗n′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 population of excited vibrational sublevels of both the reactant and product states. However vibrational sublevels of the reactant state may be omitted if the DACs are excited into the red edge of the charge transfer band. Figure 1 shows several excited vibrational sublevels of the product state in the case M = 1. Here a single-quantum mechanism of high-frequency mode relaxation and the transitions (n )

nα → nα − 1 proceed with the rate constant 1/τvαα are assumed. In Eqs. (4) – (5) the Smoluchowski operators Lˆ P and Lˆ R , correspondingly ( 2 ) 1 ∂ 2 ∂ ∑ 1 + Qi ∂ Qi + ⟨Qi ⟩ ∂ Q2 i=1 τi i ( 2 2 ) 1 ∂ 2 ∂ = ∑ + ⟨Qi ⟩ 2 1 + (Qi − 2Eri ) τ ∂ Q ∂ Qi i i i=1 2

Lˆ P =

(6)

Lˆ R

(7)



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(⃗n)

describe the diffusion on the free energy surfaces of the product, UP , and reactant, UR , states (Qi − 2Erm )2 + ∑ nα h¯ Ωα + ∆GCR 4Eri α i=1 2

UP = ∑ (⃗n)

(8)

2

Q2i i=1 4Eri

UR = ∑

(9)

where ∆GCR is the CR free energy change and Qi (i = 1, 2) are the reaction coordinates corresponding to different solvent relaxation modes with the reorganization energy Eri = xi Erm and relaxation timescale τi , 59 Erm is the solvent reorganization energy. In the base of this model lies the concept of parabolic free energy surfaces constructed in the space of solvent polarization coordinates. 1 Motion of particles along these surfaces reflects the reorganization of a solvent in the course of charge transfer and is fully characterized by the solvent relaxation function 60–63 2

X(t) = ∑ xi e−t/τi

(10)

i=1

where xi is the weight of ith relaxation mode. The dispersion of the equilibrium distribution along the ith reaction coordinate (i = 1, 2) is equal to ⟨Q2i ⟩ = 2Eri kB T . Electron transitions between the reactant state, UR , and vibrational sublevels of the product (⃗n)

state, UP , are described by the parameters ) 2π Vel2 F⃗n ( (⃗n) k⃗n = δ UP −UR , h¯

Sαnα e−Sα F⃗n = ∏ nα ! α

(11)

where F⃗n is the Franck-Condon factor, Sα = Ervα /¯hΩα and Ervα are the Huang-Rhys factor and the reorganization energy of the α th high-frequency vibrational mode, respectively. Here we use a universal spectral density of high-frequency vibrational modes for the DACs as the CR rate was shown previously 56 to depend weakly on the vibrational spectral density provided the total reorganization energy, Erv = ∑α Ervα , is constant and the number of the high-frequency vibrational modes, M, is fixed. As the universal spectral density we accept the high-frequency vibrational spectr of the DAC consisting of phenylcyclopropane (PhCP) as the electron donor and tetracya6 ACS Paragon Plus Environment

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noethylene (TCNE) as the electron acceptor. 56,64 The values of spectral parameters of the complex PhCP-TCNE are given in Table 1. Table 1: Parameters of high-frequency vibrational modes for CR in excited DAC consisting of phenylcyclopropane (PhCP) as electron donor and tetracyanoethylene (TCNE) as electron acceptor 64

α =1 h¯ Ωα , eV 0.1272 Sα 0.1245

2 3 4 5 0.1469 0.1823 0.1935 0.1993 0.1211 0.1143 0.5150 0.2302

To specify the initial conditions we assume that the system is initially in the ground state with the nuclear coordinates distributed according to the Boltzmann law (area A in Figure 1). A short laser pulse with a duration, τe , that is much shorter than the relaxation time of the solvent,

τi (i = 1, 2), vertically transports the system in the charge separated state (area B in Figure 1). In Figure 1 the DAC excited state with a nonequilibrium nuclear configuration is visualized as a wave packet located on the term UR . Assuming that only the ground state of intramolecular high-frequency vibrational mode is populated, the following expression for the initial probability distribution function on the reactant term can be written [ ] (Qi − 2Eri )2 ρR (Q1 , Q2 ,t = 0) = ∏ √ exp − 2π ⟨Q2i ⟩ i=1 2π ⟨Q2 ⟩ 2

1

(12)

i

A series of relaxation processes and electronic transitions occurs after photoexcitation is shown schematically in Figure 1. The movement of the wave packet toward the term minimum, UR , reflects the reorganization of the solvent (the blue solid arrow). The electronic transitions during (⃗n)

the wave packet relaxation occurring at the intersection points of the terms UP and UR are called nonequilibrium or hot CR (the black curved arrows). The transitions result in population of vibrationally excited sublevels of the ground electronic state (the black dashed parabolas). The vibrational excited states relax according to single-quantum mechanism, nα → nα − 1, to their ground state (the short red vertical arrows). After thermalization of the wave packet the CR can proceed 7 ACS Paragon Plus Environment


U

U

U UU U (3)

R

P

B

pump

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C

(2)

P

(1)

P

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(0)

P

- G

CR

A

2E

rm

Q

Figure 1: Multichannel nonequilibrium CR in DACs resulting in excitation of intramolecular highfrequency vibrations. The free energy curves corresponding to excited vibrational states of the product are shown as dashed lines. The initial reactant state distribution for nonequilibrium CR is pictured by a blue line, for thermal ET is pictured by a red line. in the thermal regime. The system of Eqs. (4) – (5) with the initial condition Eq. (12) is solved numerically using the Brownian simulation method. 58,65 In the numerical solution the normalized population of the charge separated state of DAC, PCS , and the effective rate constant of CR, kCR , are determined as ∫

PCS (t) = −1 kCR

ρR (Q1 , Q2 ,t)dQ1 dQ2

(13)

∫ ∞

=

PCS (t)dt

0

(14)

The model accounts for the local reversibility of electron transfer that can be adequately described only if the intramolecular vibrational relaxation or the vibrational redistribution is also taken into consideration. The intramolecular vibrational redistribution is well known to proceed (1)

on the timescale of τv ∼ 100 fs. 66 In simulations the value τvα = 150 fs is set for all vibrational modes. 67


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Results and Discussion In the simulations the dynamical parameters of real solvents such as acetonitrile (ACN) and valeronitrile are used. The solvent relaxation is described by two relaxation timescales, τi , i = 1, 2. 60–63 These timescales are attributed to different relaxation modes corresponding to distinct types of motions of the solvent molecules. The first time, τ1 , is associated with the fast inertial relaxation, and the second, τ2 (τ1 < τ2 ), with the slower diffusion relaxation. The time τ1 is almost independent of the medium viscosity, while τ2 is proportional to the viscosity. Therefore the DSE should be largely attributed to the influence of diffusion relaxation component, τ2 . The results of numerical simulation of CR kinetics in photoexcited DACs employing the multichannel stochastic model are presented in Figure 2, Figure 3, and Figure 4. At first we analyze the −1 , dynamical solvent effect on CR kinetics, namely a dependence of the effective CR time, τeff = kCR

on the solvent relaxation time, τ2 , presented in Figure 2. Hereinafter the dynamic parameters of the solvent (weights and relaxation timescales) are equal to x1 = 0.2, x2 = 0.8 (frames A–C), x1 = x2 = 0.5 (frames D–F); the fast inertial relaxation time is fixed, τ1 = 0.19 ps, while the second time, τ2 is varied from 0.50 ps (ACN) 44 to 10 ps. Variable parameters of the stochastic model in the calculations are the electronic coupling parameter, Vel , the free energy gap, -∆GCR , the intramolecular Erv and the solvent Erm reorganization energies. The analysis of the data presented in Figure 2 allows formulating five trends: (i) the effective CR time monotonically increases with rise of the solvent relaxation time, τ2 , that is, DSE is predicted for the all region of the parameters considered, (ii) the DSE is most pronounced in the region of strong exergonic CR, ∆GCR = −1.5 eV (frames A and D) and weakens with decreasing the CR exergonicity (frames C and F), (iii) the magnitude of DSE (an effective slope of the curve picturing the dependence of τeff on τ2 ) decreases with the growth of the electronic coupling value, (iv) with a rise of the reorganization energy of the high-frequency vibrational modes, Erv , the DSE attenuates (compare the left half and the right panels in Figure 2), (v) increasing the weight of slow solvent relaxation mode, x2 , enhances the DSE (compare dashed and solid lines in Figure 2). The first and fifth trends are similar while the second and third are opposite to that observed in the 9 ACS Paragon Plus Environment


E (A)

G

E

=0.8 eV, E =0.2 eV

rm

20

=0.5 eV, E =0.5 eV

rm

rv

rv

8

=-1.5 eV

CR

(D)

G

=-1.5 eV

CR

0.02

, ps

16 0.02

eff

12

0.05

6 0.02

eff

, ps

0.02

4

0.09

0.05

8 0.05

0.09

2

0.09

4

0.05 0.09

0

0 0

2

4

6

8

0

10

2,0

8

2

(E)

=-1 eV

CR

G

4

6

8

6

0.02

1,5

eff

0.05

10

=-1 eV

CR

0.02

, ps

G

eff

, ps

(B)

0.02

1,0

0.09

4

0.05

0.02

2

0,5

0.05

0.05 0.09 0.09

0.09

0,0

0 0

2,5

2

(C)

G

4

6

8

0

10

(F)

=-0.5 eV

CR

2

G

4

6

8

=-0.5 eV 0.02

0,4

1,5

0.05

, ps

2,0

0.02

0,3

eff

, ps

10

CR

0.02

eff

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0,2

1,0

0.09

0.05 0.02

0,5

0,1

0.05

0.05

0.09

0

2

4

6

8

0.09 0.09

0,0 0

10

, ps

2

4

6

8

10

, ps

2

2

Figure 2: Dependence of CR effective time, τeff , on the slow solvent relaxation time, τ2 . The left and right halves of the figure show the data obtained with values of the reorganization energies Erm = 0.8 eV and Erv = 0.2 eV (frames A – C), Erm = 0.5 eV and Erv = 0.5 eV (D – F). The values of the free energy gap, −∆GCR in eV, are indicated on the frames. The weights of the solvent relaxation modes are x1 = x2 = 0.5 (solid lines) and x1 = 0.2, x2 = 0.8 (dashed lines). The values of the electronic coupling, Vel (in eV), are placed near the lines.


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E

=0.8 eV

E

rm

=0.5 eV

rm

P

(A)

G

P

=-1.5 eV

CR

CS

(D)

CS

0

G

=-1.5 eV

CS

0

10

10

-1

-1

10

10

0.5 0.5

5

0.5

5

-2

5

0.5

5

-2

10

10

0

5

10

(B)

P

G

15

20

25

0

=-1 eV

2

P

CR

4

(E)

CS

CS

6

G

8

10

12

=-1 eV

CR

0

0

10

10

-1

-1

10

10

0.5

0.5

0.5

5

5

0.5

5

5

-2

10

-2

0

1

P

2

(C)

CS

3

G

4

5

10

6

0,0

0,5

P

=-0.5 eV

(F)

CS

CR

0

1,0

G

1,5

=-0.5 eV

CR

0

10

10

-1

-1

10

10

0.5

5

5

0.5

0.5

5

0.5 5

-2

-2

10

10

0

2

4

6

0,0

8

t, ps

0,5

1,0

1,5

2,0

2,5

t, ps

Figure 3: The time dependence of the excited-state population kinetics. The left and right halves of the figure show the data obtained with values of the reorganization energies Erm = 0.8 eV and Erv = 0.2 eV (frames A – C), Erm = 0.5 eV and Erv = 0.5 eV (D – F). The values of the free energy gap, −∆GCR in eV, are indicated on the frames. The electronic coupling Vel = 0.02 eV (solid lines) and Vel = 0.07 eV (dashed lines). The values of the solvent relaxation time, τ2 (in ps), are placed near the lines.



E

=0.8 eV, E =0.2 eV

rm

E

rv

=-1.5 eV

-1

, ps

5

2 1

eff

-1

, ps

0.5

1 1

1

k

eff

k

0.5

0.5

0,6 0,4

5

1

5

0,2

5

0,0

0 0,02

0,04

0,06

0,08

0,10

0,12

0,02

12 (B)

G

=-1 ev

0,04

(E)

0.5

CR

3,0

0,06

G

0,08

0,10

=-1 eV

CR

0,12

0.5

10

k

eff

0.5 5

8

1

eff

1

2,0

1

-1

, ps

0.5

2,5

-1

=-1.5 eV

CR

3

0,8

, ps

G

rv

0.5

CR

1,0

k

=0.5 eV, E =0.5 eV

rm

(D) G

(A)

1,2

1

1,5

6 5

5

4

5

2

1,0 0,5

0

0,0 0,02

0,04

0,06

0,08

0,10

G

=-0.5 eV

0,06

0,08

0,10

0,12

G

=-0.5 eV

0.5

CR

1

5

5 0.5

-1

1 5

, ps

0.5

1

30

eff

eff

-1

, ps

0,04

(F)

40

CR

5

3

0,02

0.5

(C)

6

4

0,12

1

k

7

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

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20

5

2

10 1 0

0 0,02

0,04

0,06

0,08

0,10

0,12

0,02

V , eV

0,04

0,06

0,08

0,10

0,12

V , eV

el

el

Figure 4: Dependence of CR kinetics rate constant, kCR , on the electronic coupling, Vel . The left and right halves of the figure show the data obtained with values of the reorganization energies Erm = 0.8 eV and Erv = 0.2 eV (frames A – C), Erm = 0.5 eV and Erv = 0.5 eV (D – F). The values of the free energy gap, −∆GCR in eV, are indicated on the frames. The weights of the solvent relaxation modes are x1 = x2 = 0.5 (solid lines) and x1 = 0.2, x2 = 0.8 (dashed lines). The values of the solvent relaxation time, τ2 (in ps), are placed near the lines.


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ET thermal reactions. The kinetics of the thermal ET are not simulated in this paper because the regularities inherent to such reactions were analyzed in detail in refs 38,39. The fourth trend did not explicitly discuss in the literature although it follows from the analysis of decoupling of solvent and ET dynamics presented in ref 68. The opposite dependencies of DSE for thermal and nonequilibrium ET rates are due to different initial positions of the wave packets on the reactant term (red and blue wave packets in Figure 5). The thermal wave packet is located always at the bottom of the reactant term while the nonequilibrium packet initially is far away from the term minimum. In the area of moderate exergonicity (Figure 5 A) the thermal wave packet locates in the region of sufficiently effective sinks (point B) while the nonequilibrium packet is initially localized in the region of much weaker sinks. In the second case, delivery of particles to the reaction zone is required, but not in the first. The delivery stage is clearly seen as a quasi-plateau in the short time domain in Figure 3 where time dependencies of the reactant state population, PCS (t), are shown. The quasi-plateau is most clearly seen in the frames A and D (blue solid and dashed lines). It can be seen that the delivery time depends on the solvent relaxation time, τ2 , the electronic coupling, Vel and the free energy gap, -∆GCR . The delivery time dependence on the solvent relaxation time, τ2 , is a reason of the influence of the dynamic solvent characteristics on the effective CR time. This results in strong enough DSE for nonequilibrium CR and weak DSE for thermal reactions in the region of moderate exergonicity. For weakly exergonic CR, −∆GCR < Erm + Erv , ( Figure 5 B) the picture is opposite. Namely, the nonequilibrium wave packet immediately appears in the reaction zone and the thermal particles have to diffuse in order to achieve the reaction zone. As a result the DSE is weak or completely absent for nonequilibrium CR (frames C and F in Figure 2). Such behavior of DSE in non-equilibrium CR is just opposite to thermal ET reactions. Indeed, theoretical investigations of the DSE in thermal ET have shown that it is strong in the range of weak exergonicity and it disappears in the range of strong exergonicity. 38,39 It should be stressed that these conclusions are valid only when the reorganization energy of the high-frequency vibrational modes, Erv , is large enough.



(A)

(B)

ffu di

free energy

sio

n

reactant Free energy

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A

B

reactant A n

io us

ff di

B

product

product

reaction coordinate

reaction coordinate

Figure 5: Schematic representation of the free energy curves for the multichannel CR reaction. A location of the CR reaction zone in the strong (panel A) and weak (panel B) exergonic regions are presented. The free energy curves corresponding to excited vibrational states of the product are shown as dashed lines. The initial reactant state distribution for nonequilibrium CR is pictured by a blue line, for thermal ET is pictured by a red line. Addressing to the third trend, we note that in the thermal reactions the DSE value monotonically increases with electronic coupling increase achieving saturation at large Vel . Indeed, Eq. (1) shows that in the nonadiabatic limit, g ≪ 1, the rate constant is independent of the solvent relaxation time, τL , and proportional to Vel2 . In the opposite limit (solvent controlled regime), g ≫ 1, the rate constant is in inverse proportion to the solvent relaxation time, τL , and independent of Vel2 . CR in excited DACs proceeds mainly in nonequilibrium mode, during the wave packet motion from its initial position to the term minimum. In this case the sink effectiveness is characterized by the probability of electronic transition, WNET , during the wave packet passing through a term crossing 48,69

where

2π Vel2 τL WNET = h¯ (1 + gNET )|∆GCR − Erm |

(15)

[ ] 2π Vel2 τL 1 1 gNET = + h¯ |∆GCR + Erm | |∆GCR − Erm | fv

(16)


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is the reaction nonadiabaticity parameter, which depends on the vibrational relaxation factor [ fv = 1 +

8Erm kB T τL (∆GCR − Erm )2 τv

]1/2 (17)

Here τv is the relaxation time of the high-frequency vibrational mode. The vibrational relaxation factor, fv , varies from 1 to ∞ when the vibrational relaxation time, τv , changes from ∞ to 0. Eq. (15) is applicable when the condition |∆GCR − Erm | ≫ kB T is met. 69 The probability WNET is proportional to the solvent relaxation time, τL , and Vel2 in the limit of weak electronic coupling, gNET ≪ 1, and in the limit of strong electronic coupling, gNET ≫ 1, the probability is independent of Vel and τL . These regularities explain the third trend. The results displayed in the Figure 2 (compare dashed and solid lines) also show a growth of the effective slope of the curve picturing the dependence of τeff on τ2 with an increase of the weight of the slow relaxation component, x2 (a variation of x2 must conserve the condition x1 + x2 = 1). This is an obvious trend reflecting a grows of τeff with increase of the effective relaxation time of the solvent due to increase of x2 . The trend becomes stronger with the decrease of the electronic coupling. Figure 4 demonstrates the results of numerical calculations of the CR rate constant, kCR , as a function of the electronic coupling, Vel . Here the variable model parameters are the second solvent relaxation time, τ2 , the free energy gap, -∆GCR , the energies intramolecular, Erv , and the solvent, Erm , reorganization. In the calculations Vel is varied in the range from 0.02 eV to 0.12 eV, because the best fitting to the experimental data on the CR rate constant dependence on the free energy gap 41–44 was obtained with the values from this area. 52,53 The results of numerical simulations, Figure 4, showcase a monotonic growth of the CR rate constant, kCR , with increasing Vel . However, in all cases this dependence is substantially weaker than that expected in nonadiabatic regime where it is quadratic. This clearly indicates a significant role of the DSE. With increasing the reorganization energy of the high-frequency vibrational modes, Erv , the dependence of kCR on Vel enhances (compare left and right frames). This trend is



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easily explained. With increasing Erv the power of individual sinks becomes smaller, the reaction is closing to the nonadiabatic regime where the dependence of kCR on Vel is stronger. The dependencies of kCR (Vel ) pictured in frames A – C clearly show approaching to its saturation. This is a consequence of the dependence saturation predicting for both the thermal and nonequilibrium reactions (see Eq. (1) and Eq. (15)) in the solvent controlled regime. Another regularity is the strengthening of the dependence of kCR (Vel ) with the rise of CR rate constant. The regularity is particularly pronounced in the case of Erv =0.5 eV (frames D–F). It is quite unexpected because this is an opposite conclusion to that following from Eq. (1) for the thermal ET rate constant. Indeed, faster reaction supposes more powerful sinks and, hence, larger √ (⃗n) (⃗n) values of effective electronic coupling, Veff = Vel F⃗n , but in the area of large Veff the dependence kCR (Vel ) is saturated. To understand this regularity we note that for ∆GCR =-1.5 eV (i) the reaction essentially proceeds in nonequilibrium regime during the wave packet relaxation because for all parameters the relation is met τe f f ∼ τ2 , (ii) in the course of the relaxation the wave packet moves from the weakest sinks to stronger (the most powerful sink is placed to the left from point C in Figure 1). The increase of electronic coupling, Vel , enhances the weak sinks and the reaction zone shifts in the direction from the point C to B (see Figure 1), as a result, the reaction proceeds through the sinks (⃗n)

approximately with the same values of the effective electronic coupling, Veff , but with different ⃗n. This weakens the dependence of kCR (Vel ). In contrast, for ∆GCR =-0.5 eV the most powerful sink is placed between the points C and B in Figure 1. In this case the highest sinks being characterized by a large effective electronic coupling value have minor effectiveness due to small term slope of the product state. Indeed, from Eq. (15) it follows that the maximum value of nonequilibrium ET probability achieving in the solvent controlled regime is (⃗n)

WNET =

|AP | (⃗n)

AR + |AP |

(⃗n)

(⃗n)

(18)

where AR and AP are the slopes of terms UP and UP at the point of their intersection, respectively.


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(⃗n)

For the highest sinks the inequality is held AR ≫ |AP | that results in minor WNET . So that the reaction mainly occurs in the area where the sink power goes down with approaching to the point C. The increase of electronic coupling, Vel , enhances the sink power and the reaction zone again shifts in the direction from the point C to B (see Figure 1) and now the reaction proceeds through (⃗n)

the sinks with much larger values of the effective electronic coupling, Veff . This considerably strengthens the dependence of kCR (Vel ) and leads to almost quadratic dependence of kCR on Vel for weak exergonic CR (frame F in Figure 4). Time dependencies of the population decay of the photoexcited state shown in Figure 3 demonstrate the extent of non-exponentiality of the CR kinetics. The non-exponentiality is most pronounced for relatively slow reactions. In the experiments only small non-exponentiality was observed that can be described in terms of a stretch exponential function, 49,70 so that we can conclude that the real DACs are typically characterized by rather strong electronic coupling. Two-staged regime most clearly manifests if the solvent reorganization energy noticeably exceeds the CR free energy gap (black solid line in frame C in Figure 3). Otherwise the CR kinetics closely approach to the single nonequilibrium stage regime for strong and intermediary exergonic CR and with rise of electronic coupling. 57

Conclusions The study of ultrafast ET kinetics in the framework of the multichannel stochastic model allowed us to identify similarities and differences in trends that are inherent to charge transfer reactions occurring in the nonequilibrium and equilibrium regimes. As a specific process the charge recombination in excited DACs is studied. The main conclusions can be summarized as follows. 1. DSE is predicted for both the nonequilibrium and equilibrium charge transfer. But there is an important difference. For thermal reactions DSE is strong in the area of weak exergonicity and it is weak in the area of strong exergonicity while for the nonequilibrium reactions the regions of strong and weak DSE are in the opposite areas. 17 ACS Paragon Plus Environment


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2. Increasing the electronic coupling value results in decrease of DSE magnitude for nonequilibrium ET and in its growth for thermal reactions. 3. The nonequilibrium CT kinetics can considerably deviate from the exponential law but with increasing the electronic coupling the kinetics approach to the exponential regime. 4. In the area of weak exergonicity the CR kinetics can be biphasic corresponding to two stages, fast nonequilibrium and equilibrium that is considerable slower. The biphasic kinetics can be observed in DACs with weak electronic coupling and in fast solvents.

Acknowledgement The study was performed by a grant from the Russian Science Foundation (Grant No. 16-1310122).

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weak exergonic ET

ffu sio di

B

product

n

reactant

A

Free energy

strong exergonic ET Free energy

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reaction coordinate diffusion is necessary only for nonequlibrium ET

reactant

product A

n sio ffu di B

reaction coordinate diffusion is necessary only for thermal ET


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Dynamic Solvent Effect on Ultrafast Charge Recombination Kinetics in

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