Free-Energy-Gap Law for Ultrafast Charge Recombination of Ion Pairs

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Free Energy Gap Law for Ultrafast Charge Recombination of Ion Pairs Formed by Intramolecular Photoinduced Electron Transfer Alexey E Nazarov, Roman E. Malykhin, and Anatoly Ivanovich Ivanov J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b10550 • Publication Date (Web): 01 Jan 2017 Downloaded from http://pubs.acs.org on January 5, 2017

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The Journal of Physical Chemistry

Free Energy Gap Law for Ultrafast Charge Recombination of Ion Pairs Formed by Intramolecular Photoinduced electron Transfer Alexey E. Nazarov, Roman Malykhin, and Anatoly I. Ivanov∗ Volgograd State University, University Avenue 100, Volgograd 400062, Russia E-mail: [email protected]

1 ACS Paragon Plus Environment


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Abstract The regularities of ultrafast charge recombination kinetics in photoinduced intramolecular electron transfer in polar solvent are studied. The kinetics of charge separation and ensuing ultrafast charge recombination are simulated within the framework of the multichannel stochastic model. The model accounts for the reorganization of both the solvent and a number of the intramolecular high-frequency vibrational modes. The solvent relaxation is described in the terms of two relaxation modes.For ultrafast charge recombination the free energy gap law strongly depends on the parameters: the electronic coupling, reorganization energy of intramolecular high-frequency vibrational modes, the vibrational and solvent relaxation time. The semilog dependence of the charge recombination rate constant on the free energy gap varies from the parabolic shape to a nearly linear one with increasing the electronic coupling and decreasing the vibrational relaxation time. The dynamic solvent effect in charge recombination is predicted to be large in the area of strong exergonicity and to be small in the area of weak exergonicity. This regularity is opposite to that observed for the thermal reactions.

Introduction Light induced charge separation is one of the most common processes when photoexcitation is exploited as driving force for photochemical transformations. Photoinduced electron transfer is thus of paramount importance for both fundamental science and technological applications. 1–7 Such processes often have low efficiency because of short lifetime of the charge separated states that considerably inhibits a quantum yield of desired products. Typically back electron transfer (BET) is responsible for the decay of generated ion pairs in intermolecular photoinduced electron transfer reactions. Hence it is essential to establish strategies to achieve long-lived photoinduced charge separated states. The main obstacle is that the dependencies of the ultrafast BET rate constants on many controlling parameters are still unknown because a detailed mechanism of BET reactions is mostly not understood in spite 2 ACS Paragon Plus Environment

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of its importance for control of photochemical process effectiveness. 7 The essential feature of ultrafast charge transfer mechanism is a strong nonequilibrium of the nuclear subsystem. The regularities of such reactions may be significantly different from those observed for the thermal reactions. 8 The free energy gap (FEG) is the most common parameter controlling the charge transfer rate. The dependence of the rate on the FEG known as the Marcus FEG law predicts the electron transfer (ET) to be slow in either the region of strong or weak exergonicity but fast in between, reaching the maximum at −∆G = Erm . 9 Such a bell-shaped free energy dependence was experimentally confirmed for several kinds of charge transfer. The intramolecular charge shift was the first among them. 10,11 Later, it was observed for many other types of ET. 12–20 One of the reactions which demonstrates non-Marcus dependence of the rate constant on the reaction free energy is the geminate charge recombination (CR) in donor-acceptor complexes (DAC) excited into the charge transfer band. 21–23 For this reaction the Marcus normal region is absent and the logarithm of the rate constant decreases nearly linearly with increasing the reaction exergonicity. To account for such a behavior of the CR rate constant Mataga and coworkers suggested that the CR in the excited DACs is associated with reorganization of intramolecular vibrational high-frequency modes and the solvent plays a minor role. 22 Fundamentally different explanation was proposed by Tachiya and Murata. 24 They took into account that a short laser pulse populates a non-equilibrium initial vibrational state of a DAC. However the explanation encountered two problems: 25 (i) a good fit required too large electronic coupling values and (ii) the model predicted a strong time dependence of the rate constant, in disagreement with most experimental data. 25 Later it was shown that both problems are connected with oversimplification of the model due to full neglect of the reorganization of intramolecular high-frequency modes. 26 Indeed, a rather good fitting of the available experimental data on the FEG dependence of the CR rate constant in excited DACs was obtained within multichannel stochastic point-transition model including the re-



(A)

S

excitation

1

CSS

S

0

Free energy

GCR

(B)

excitation

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+

-

D A GCR

DA

Reaction coordinate

Figure 1: Cuts in the free energy surface of the ground, first excited, and the charge separated state. (A) photoinduced forward ET and ensuing back ET and (B) CR in excited DACs. The dashed lines are the vibrational repetitions of the electronic terms. The repetitions are equidistant in the Figure, but for the model with several vibrational modes the density of the sublevels sharply increases with the free energy rise. Electronic transitions occur at the intersections of all sublevels and shown with red curved arrows. The magenta arrows stand for the solvent relaxation. The relaxation of the high-frequency mode is shown by the green vertical arrows as transitions between the neighbor vibrational sublevels. organization of the solvent and several intramolecular high-frequency vibrational modes. 26 At the same time the CR kinetics appear to be close to exponential. 27 The multichannel stochastic point-transition model is a combination and generalization of the Bixon-Jortner and Sumi-Marcus models. 28,29 The Bixon-Jortner model is generalized to include reorganization of several high frequency vibrational modes and go beyond the Golden Rule limit. The Sumi-Marcus model involves two classical relaxation modes with strongly different relaxation time constants whereas the multichannel stochastic model operates with several classical modes with arbitrary ratio of their relaxation time constants. In this work we highlight a very deep analogy between the CR of the excited DACs and


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the back ET (A∗ D → (forward ET) A− D+ → (back ET) AD) in intramolecular photoinduced ET reactions (compare frames A and B in Figure 1). Indeed, both reactions occur in essentially non-equilibrium mode, parallel to the relaxation of the nuclear subsystem. This leads to a lot of common features that can dramatically differ from those observed for the thermal reactions. The influence of a number of the parameters (electronic coupling, solvent relaxation time constant, reorganization energy of the solvent and intramolecular high frequency vibrational modes) on ultrafast CR kinetics in excited DACs was studied and reported in ref. 30. Besides the deviation of the kinetics from exponential law was discussed in details. In CR ensuing photoinduced charge separation the Marcus normal region (MNR) also was not observed. 12,31–40 At the moment it is commonly accepted that MNR in intramolecular CR is not observed because there are no donor-acceptor pairs with small enough FEG. However, this is true only for the photoinduced charge separation from the first excited state when the recombination proceeds to populate the ground state of the donor-acceptor pair. Currently, studies of the kinetics of charge transfer from the second excited state become available. 18,41–52 In such processes CR in the first excited state is possible. In this case the FEG can be small and moreover the recombination free energy change can be even positive. There is strong evidence that the CR even in the region of the positive recombination free energy occurs in ultrafast mode. 45,46,51,53,54 This, in its turn, evidences that the MNR is also either absent or strongly suppressed in intramolecular ultrafast back electron transfer. The main aims of the paper are (i) to develop a model of ultrafast CR in photoinduced intramolecular CS (ii) to simulate FEG law for CR proceeding in nonequilibrium regime, (iii) to investigate the effect of the solvent and vibrational relaxation times, electronic coupling and the reorganization energy of high frequency vibrational modes on the FEG law. An important question to be answered is: what is the reason of the MNR suppression in intramolecular ultrafast CR ensuing photoinduced charge separation?



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Theory and Computational Details The model considered here includes: three electronic states, namely, the ground state (GS), a locally excited state (LES), a charge separated state (CSS), and their vibrational repetitions, interaction of the transferred charge with a polar solvent; intramolecular electron-vibrational interaction. The elements of the model are presented in Figure 1 A. In real polar solvents that are typically characterized by a nonexponential relaxation function, X(t), 55–59 that can be represented by a sum of a few (N ) exponential summands: 56,57,60

X(t) =

N ∑

N ∑

xi e−t/τi

i=1

(1)

xi = 1

i=1

Each summand is associated with a separate collective solvent coordinate Qi with relaxation time constant τi . The solvent coordinates form a N -dimensional configuration space of the problem. Charge transfer can be described in terms of multidimensional free energy surfaces associating with different electronic and vibronic states participating in the reaction. The profiles of the free energy in terms of the reaction coordinate Q are pictured in Figure 1 A. (⃗l)

(⃗ n)

The diabatic free energy surfaces for the ground state, UGS , locally excited state ULES , and (m) ⃗

the charge separated electronic state UCSS with their vibrationally excited sublevels can be written as follows 61 (⃗l) UGS

(

N √ 1∑ = Qi − 2Eriex 2 i=1

(⃗ n)

ULES = (m) ⃗ UCSS

)2

+ ∆GGS +

M ∑

lα h ¯ Ωα

N M ∑ 1∑ Q2i + nα h ¯ Ωα 2 i=1 α=1

(

√ N 1∑ Qi − 2EriCS = 2 i=1

(2)

α=1

(3)

)2

+

M ∑

mα h ¯ Ωα + ∆GCS

(4)

α=1

where Ωα , nα , mα , lα (nα , mα , and lα = 0, 1, 2, ...) are the frequency and the quantum numbers of the αth quantum intramolecular vibrational mode, correspondingly, the indexes ⃗n, m, ⃗ ⃗l stand for the sets of quantum numbers ⃗n = {n1 , n2 , ...nα , ...}, m ⃗ = {m1 , m2 , ...mα , ...},


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and ⃗l = {l1 , l2 , ...lα , ...}, M is the number of the intramolecular quantum vibrational modes, their frequencies Ωα must satisfy the conditions h ¯ Ωα ≫ kB T with kB and T being the Boltzmann constant and the temperature, ∆GGS is the free energy change for electronic transition from the locally excited to the ground state, ∆GCS is the free energy change of the photoinduced charge separation, Eriex and EriCS are the reorganization energies of the i-th medium mode of the solvent at the stages of pump and charge separation, correspondingly. ex They are related to their weights in the relaxation function eq 1 as follows: xi = Eriex /Erm = CS ex EriCS /Erm , Erm =

∑ i

CS Eriex and Erm =

∑ i

EriCS are the total reorganization energies of the

solvent at the corresponding stage. There is a difference between the abbreviations CS and CSS. The abbreviation CS stands for the charge separation stage while the index CSS is used to indicate the charge separated state. In the framework of the stochastic point-transition approach, 27,61,62 the temporal evolution of the system is described by a set of equations for the probability distribution func(m) ⃗

(⃗ n)

tions for the locally excited, ρLES (Q, t), the charge separated, ρCSS (Q, t), and the ground, (⃗l)

ρGS (Q, t), states (⃗ n′ )

) ∑ ρ α ∑ (e) ( (⃗n) ∑ ρLES ∂ρLES (m) ⃗ n) LES ˆ LES ρ(⃗ − ρ − k ρ = L − CSS + LES LES ⃗ nm ⃗ (n +1)) (nα ) α ∂t α τv α τv m ⃗

(5)

) ∑ ∑ (e) ( (⃗n) ∂ρCSS (m) ⃗ (m) ⃗ (g) (⃗l) m) ⃗ ˆ CSS ρ(CSS − ρCSS + km ρ + k⃗nm = L GS ⃗ ⃗ ρLES − ρCSS + ⃗ l ∂t ⃗ ⃗ n

(6)

(⃗ n)

(⃗ n)

)

(

(m) ⃗

l

+ (⃗l)

m ⃗ ′α ) ∑ ρ(CSS (mα +1) α τv

−

m) ⃗ ∑ ρ(CSS (mα ) α τv

)

(

(⃗l′ )

(⃗l)

α ∑ ρGS ∑ (g) ∑ ρGS ⃗l) ∂ρGS (⃗l) (m) ⃗ ˆ GS ρ(GS + − ρ = L − km ρ − GS CSS (lα +1) (lα ) ⃗ l˙ ∂t α τv α τv m ⃗

(7)

⃗ ′α , ⃗lα′ are defined as where Q is a vector with components Q1 , Q2 , ..., QN , the vectors ⃗n′α , m ⃗ ′α = {m1 , m2 , ...mα + 1, ...}, ⃗lα′ = {l1 , l2 , ...lα + 1, ...}. follows ⃗n′α = {n1 , n2 , ...nα + 1, ...}, m (⃗ n) (⃗l) ˆ LES ,L ˆ CSS and L ˆ GS are the Smoluchowski operators describing diffusion on the ULES L , UGS ,



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(m) ⃗

and UCSS free energy surfaces, correspondingly, ˆ LES = L ˆ CSS L

[

N ∑ 1 i=1 N ∑

τi

[

∂ ∂2 1 + Qi + kB T ∂Qi ∂Q2i (

]

(8)

)

√ ∂ 1 ∂2 CS = 1 + Qi − 2Eri + kB T ∂Qi ∂Q2i i=1 τi

ˆ GS = L

N ∑ 1 i=1 τi

[

(

1 + Qi −

)

√

2EriGS

∂ ∂2 + kB T ∂Qi ∂Q2i

]

(9)

]

(10)

The model, eqs 5 – 7, includes single-quantum irreversible vibrational relaxation nα → nα −1 (nα ) where 63 that is supposed to proceed with the rate constant 1/τvα

(nα ) (1) τvα = τvα /nα

(11)

The intramolecular vibrational redistribution is known to proceed on the time scale of ∼ 100 fs. 64,65 (⃗ n)

Transitions between vibrational sublevels of the locally excited electronic state, ULES , (⃗l)

(m) ⃗

CSS, UCSS , and the ground state, UGS , are described by the Zusman parameters 27,61,62 ( ) 2πVe2 F⃗nm (⃗ n) (m) ⃗ ⃗ = δ ULES − UCSS , h ¯

(i)

F⃗nm ⃗ =

∏ α

{

}



(g) km ⃗ ⃗l

min(nα ,mα )



exp −Sα(i) nα !mα ! 

∑

rα =0

(

(CR)

(CS)

(e) k⃗nm ⃗

2πVg2 Fm ⃗ ⃗l = h ¯ √

δ

(m) ⃗ UCSS

−

(⃗l) UGS

,

(12)

2

(−1)mα −rα ( Sα )nα +mα −2rα   rα !(nα − rα )!(mα − rα )! (i)

)

(13)

where i = CS, CR, Ve and Vg are the electronic couplings for transitions between LES and (i)

CSS and between CSS and GS, correspondingly, F⃗nm ⃗ is the Franck-Condon factor for the transition between the vibrational sublevels ⃗n and m, ⃗ the Huang-Rhys factors Sα(i) are differCR CS /¯ hΩα /¯hΩα and SαCR = Ervα ent for charge separation and CR stages. They are SαCS = Ervα CS = with Erv

∑

CR CR CS being the reorganization energy of the intramolecular = Ervα and Erv Ervα

high-frequency modes at the charge separation and CR stages, correspondingly. The populations of all electronic states involved in the reaction are determined by the


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equations

Pk (t) =

∑∫

(⃗ n)

ρk (Q, t)

N ∏

(14)

dQi ,

i=1

⃗ n

where k = LES, CSS, or GS. The pump pulse with carrier frequency ωe is assumed to be of the Gaussian form {

}

t2 E(t) = E0 exp iωe t − 2 τe

(15)

and its duration, τe , is short so that the solvent is considered to be frozen during excitation. All high-frequency vibrational modes are supposed initially to be in the ground state. This allows obtaining the following general expression for the normalized initial probability distribution function on the locally excited term 66,67 { (⃗ n) ρLES (Q, t

= 0) = APn exp −

[¯hδωe(⃗n) −

∑ e √ Q 2E ex ]2 τ 2

2¯h

i 2

ri

e

−

e2 ∑ Q i

}

2kB T

∏ [Sαex ]nα e−Sα

(16)

ex

P⃗n =

(17)

nα !

α

ex where Sαex = Ervα /¯hΩα is the Franck-Condon factor for αth mode at the excitation stage, ∑ ex e = Q − √2E ex , P is the Franck-Condon factor h ¯ δωe(⃗n) = h ¯ ωe − Erm ¯ Ωα , Q + ∆GGS − α nα h n i i ri

at the excitation stage. The normalization factor A is determined by the equation ∫ ∑

(⃗ n) ρLES (Q, t

= 0)

n

∏

( N/2

dQi = A(2πkB T )

i

)

h ¯ Z=1 στe

(18)

ex kB T ) + h ¯ 2 τe−2 is the dispersion of an electron-vibrational transition in the where σ 2 = (2Erm

absorption spectrum,

Z=

∑

  

Z⃗n ,

Z⃗n = P⃗n exp −  

⃗ n

] 

[

2  h ¯ δωe(⃗n) 

2σ 2


 

(19)


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The probability of the population of the ⃗n-th vibrational sublevel of LES, W⃗n , is

W⃗n =

Z⃗n Z

(20)

The initial conditions for the charge separated and the ground state distribution functions, (m) ⃗

(⃗l)

ρCSS (Q, t) and ρGS (Q, t), are: (⃗l)

(m) ⃗

ρCSS (Q, t = 0) = 0, ρGS (Q, t = 0) = 0

(21)

The set of eqs 5 – 6 with the initial conditions eq 16 – 21 is solved numerically by using the Brownian simulation method. 67–69 The computer code is described in ref 70. Here we restrict ourselves to exploring only ultrafast charge separation and recombination. The fluorescence and the internal conversion from the locally excited state are not considered in this paper since they proceed much slower than the charge separation.

Results and Discussion In the beginning of this section the influence of the electronic coupling and the solvent relaxation time constant on the FEG law for the rate of the CR into the ground state is investigated. The reaction can proceed in a non-thermal regime so that its kinetics can considerably deviate from exponential. The time independent effective rate constant can strongly vary with its definition. In what follows the following determination of the effective rate constant is used 71 1 kCR

∫

= 0

t0

dtPCSS (t)

(22)

where the time moment t0 is determined by the condition PGS (t0 ) = 0.99. In the calculations the stochastic point transition model is exploited. It includes five high-frequency vibrational


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modes with the total reorganization energy Erv = 0.2 eV (the frequencies and reorganization energies of separate modes are given in Table 1). Table 1: Frequencies and weights of high-frequency vibrational modes for charge transfer in PhCP/TCNE complexes. 72

PhCP/TCNE No h ¯ Ωα , eV Ervα /Erv 1 0.1272 0.079 2 0.1469 0.089 3 0.1823 0.104 4 0.1935 0.498 5 0.1993 0.229 The magnitude of Erv is considered to be the same at the excitation, charge separation and recombination stages. The solvents with two relaxation times are exploited. Two types of solvents are considered: (i) a fast solvent with relaxation parameters of acetonitrile (ACN) which has the weights of the Debye’s modes x1 = 0.686, x2 = 0.314 and relaxation timescales τ1 = 0.089 ps, τ2 = 0.63 ps, 58 (ii) a slow solvent with the weights x1 = x2 = 0.5 and relaxation timescales τ1 = 0.2 ps and τ2 = 5.0 ps. The parameters of the slow solvent are close to those of valeronitrile, benzonitrile, and chloroform. 58,59 The total solvent reorganization energy at ex CS the excitation stage is Erm = 0.01 eV, at the charge separation stage is Erm = 0.65 eV, and

at CR stage is determined by eq 23 √ CR = Erm

√

CS − Erm

√

ex Erm

(23)

This equation directly comes from the definition of the reorganization energy which is the difference between the free energy values in a given electronic state, taken at the minimum terms of the initial and final electron states. The electronic coupling for CS is fixed Ve = 0.03 eV while that for CR, Vg , is supposed to be variable. Influence of Ve magnitude on the CR kinetics is very small.



10 (4)

1

kCR, ps-1

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(3) (2)

0.1 0.01

(1)

0.001 0.0001 -0.2 -0.4 -0.6 -0.8

-1

-1.2 -1.4

∆GCR, eV Figure 2: Dependence of the CR rate constant, kCR , on the free energy gap, ∆GCR , for several values of the electronic coupling: (1) Vg = 0.001 eV, (2) Vg = 0.005 eV, (3) Vg = 0.01 eV, (4) Vg = 0.03 eV. The stochastic simulation is pictured with signs +, the Golden Rule result ex CSS (eq 24) is presented by the solid lines. The parameters are: Erm = 0.01 eV, Erm = 0.65 eV, Erv = 0.2 eV, ∆GGS = −2.1 eV, τv = 1.0 ps, Ve = 0.03 eV, the fast solvent. The influence of the variation of the electronic coupling Vg on the FEG law for the CR rate constant in fast solvent (ACN) is shown in Figure 2. For comparison the thermal CR rate constant determined by eq 24

kth =

√ Vg2

h ¯

π

[ ]nα   [ ] CR ∑ CR ∑ ∏ e−Sα SαCR (∆GCR + Erm ¯ Ωα )2 + α nα h  exp − 

CR k T Erm B nα

α

CR k T 4Erm B

nα !

(24)

is displayed by the dashed lines. This equation is a straightforward generalization of the well-known Golden Rule expression. 28 One can see that for slow reactions when the condition kCR ≪ 1/τ2 is met the stochastic simulations and the Golden Rule expression lead to very close results. This is an expected result because the reaction is slow when the electronic coupling is small, that is, it occurs in the area of the Golden Rule applicability. In this case the non-equilibrium transitions proceeding in parallel with the nuclear relaxation play a minor role so that the reaction mainly occurs in CR CR . + Erv the thermal regime. Naturally, the CR rate constant culminates at −∆GCR = Erm


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Here the CR free energy change is ∆GCR = ∆GGS − ∆GCS . Increasing the electronic coupling, Vg , leads to raising the difference between the stochastic simulations and the Golden Rule predictions. There is at least two reasons of that. The first is the dynamic solvent effect suppressing the CR rate constant. 62 The second is caused by enhancing the efficiency of the non-equilibrium transitions that decreases the share of the ion pairs recombining in the thermal regime. Both reasons become irrelevant in the far Marcus inverted region where the nuclear subsystem must relax in order to achieve sufficiently effective sinks, which are arranged on the opposite ascending branch of the CSS free energy curve. So that the non-equilibrium CR plays a minor role. Moreover, the transitions occur in the vicinity of the term minimum 73 where the effective electronic couplings are considerably reduced by the Franck-Condon factors (see eqs 12) that approaches the electron transfer to the nonadiabatic regime. As a result, in the Marcus inverted region the difference between the stochastic simulations and the Golden Rule predictions vanishes. The difference is most CR CR pronounced in the neighborhood of the region −∆GCR = Erm +Erv where the rate constant

is maximal. In this region the CR mainly proceeds in parallel with the solvent relaxation, (⃗ n)

that is, when the wave packets, created by the charge separation event at the ULES and (m) ⃗

UCSS term intersections, move down to the minimum of the free energy curve of the CSS. This is so-called nonequilibrium or hot mode. An important reason of the CR rate constant reduction comparing with that obtained in the Golden Rule approximation is the creation of the wave packets on the CSS term far away from the most effective sinks which are placed at the bottom of the term. The CR rate constant increases during the wave packet motion achieving its maximum in the final point of the relaxation. In the thermal regime the wave packet is initially placed in the area of the most effective sinks and the rate is comparable with the maximal value of the nonequilibrium rate constant. Further decreasing of the CR FEG leads to the Marcus normal region where the thermal rate constant is an increasing function of the free energy gap, |∆GCR |. At the same time, in this region the nonequilibrium wave packet reaches the bottom of the term having passed through the most effective



100 (2)

(3)

(1)

10

kCR, ps-1

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1

0.1 -0.2 -0.4 -0.6 -0.8

-1

-1.2 -1.4

∆GCR, eV Figure 3: Influence of the solvent relaxation time on the free energy gap law. The stochastic ⊙ simulations are pictured with signs + (the fast solvent) and (the slow solvent), the Golden Rule result (eq 24) is presented by the solid lines. (1, red) Vg = 0.03 eV, (2, black) Vg = 0.05 eV, (3, blue) Vg = 0.07 eV, τv = 0.1 ps. The rest parameters are the same as in Figure 2 caption. sinks. As a result, the nonequilibrium effective rate constant decreases slower with decreasing |∆GCR | than the thermal one so that two curves approach each other and can intersect. In other words, the non-equilibrium and dynamic solvent effects in the Marcus normal region act in the opposite directions whereas in the neighborhood of the activationless region they act in parallel. It should be also noticed that in the region of small FEG the CS proceeds in the Marcus inverted region that leads to population of the excited vibrational sublevels (m) ⃗

of the CSS, UCSS . These excitations also influence on the nonequilibrium effective CR rate constant. 71 A shift of the curve maximum to the lesser values of the FEG with increasing the electronic coupling is one more important regularity shown in Figure 2. This means that the prediction of the Marcus theory and its generalization that the rate constant of a thermal reCR CR is not valid for ultrafast reactions proceeding +Erv action has a maximum at −∆GCR = Erm

in a nonequilibrium mode.


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Figure 3 shows the FEG law for fast CR and the influence of the solvent relaxation time on the rate constant. One can see that in the area of the CR rate constant maximum the difference between results of the Golden Rule and the stochastic simulations can exceed two orders of magnitude. For the parameters adopted the Marcus normal region is seen but it is strongly suppressed. The suppression degree increases with slowing down the solvent relaxation rate (increasing τ2 from 0.63 ps to 5 ps). The origin of this trend can be understood if one takes into account that CR proceeds mainly in the nonequilibrium regime, at the stage of the wave packet motion from its initial position to the term minimum of the CSS. For such a recombination the effectiveness of a sink is determined by the probability of CR produced by the wave packet passing through the sink, WCR , 74,75

WCR =

where

2πVel2 τL h ¯ (1 + gCR )|∆GCR + Erm |

[

gCR

1 2πVel2 τL 1 = + h ¯ |∆GCR + Erm | |∆GCR − Erm |fv

(25)

]

(26)

is the reaction nonadiabaticity parameter and [

8Erm kB T τL fv = 1 + (∆GCR − Erm )2 τv

]1/2

(27)

Here for simplicity the solvent with single relaxation time, τL , is considered. Equation 25 is applicable only when the condition |∆GCR − Erm | ≫ kB T is fulfilled. 74 One can see that the probability WCR is proportional to the solvent relaxation time, τL , in the limit of weak electronic coupling, gCR ≪ 1, and is independent of τL in the opposite limit, gCR ≫ 1. This result shows that the effectiveness of nonequilibrium CR increases with increasing the solvent relaxation time achieving a saturation in the limit of large τL . When a larger part of the particles recombine at the non-equilibrium stage the FEG law strongly deviates from that predicted by the Golden rule.



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The second regularity shown in Figure 3 can be formulated as increasing the dynamic solvent effect (influence of the solvent relaxation time on charge transfer rate constant) with the rise of the FEG which is the smallest in the region of low exergonicity and the largest in the region of high exergonicity (compare the distance between the signs + and the

⊙

of the

same colors). This trend is opposite to that observed for the thermal charge transfer. 73 The opposite dependencies of dynamic solvent effect on FEG for thermal electron transfer and CR occurring in nonequilibrium regime are caused by different initial positions of the wave packets on the CSS term. In thermal reactions the wave packet is initially located at the bottom of the CSS term and the nonequilibrium wave packet is created far away from the term minimum. In the area of strong exergonicity, Erm + Erv > −∆GCR , the thermal wave packet places in the region of sufficiently effective sinks while the nonequilibrium packet is initially localized in the region of much weaker sinks. As a result, for the thermal reaction the charges can recombine without diffusional delivery of the particles to the reaction zone while for nonequilibrium CR such a delivery is needed. Since the delivery time depends on the solvent relaxation time, τ2 , one can expect that in the second case the dynamic solvent effect should be considerably larger than in the first. For weakly exergonic CR, −∆GCR < Erm , the picture is inverted. Indeed, the nonequilibrium wave packet immediately appears in the area close to reaction zone and the thermal wave packet has to be delivered to the reaction zone. So that the dynamic solvent effect is weaker for nonequilibrium CR than that for the thermal CR. Figure 4 shows influence of the reorganization energy of the high-frequency vibrational modes on the free energy gap law. Increasing Erv from 0.2 to 0.4 eV weakly changes the CR rate constant in the region of weak exergonicity, −∆GCR < Erm , and considerably increases the rate in the area of strong exergonicity. This results in noticeable reduction of the CR rate constant variation in the area of exergonicity 0.2 < −∆GCR < 1.4 eV. For Erv = 0.4 eV the Marcus normal region becomes more pronounced. Two more interesting trends are: (i) a rather strong influence of Erv variation on the CR rate constant in the area of strong


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100 (1)

(2)

10

kCR, ps-1

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1

0.1 -0.2 -0.4 -0.6 -0.8

-1

-1.2 -1.4

∆GCR, eV Figure 4: Influence of the reorganization energy of the high-frequency vibrational modes on the free energy gap law. The stochastic simulations are pictured with signs + (the fast ⊙ solvent) and (the slow solvent), the Golden Rule result (eq 24) is presented by the solid lines. (1, black) Erv = 0.2 eV, (2, red) Erv = 0.4 eV, Vg = 0.07 eV, τv = 0.1 ps. The rest parameters are the same as in Figure 2 caption. exergonicity and (ii) disappearance of the influence in the area of weak exergonicity. The first trend is expected because it is well known that for the thermal reactions increasing Erv increases the rate constant in the Marcus inverted region and in this region the deviation of CR from the thermal regime is not very strong. The second trend reflects the competition of increasing the efficiency of non-thermal CR and diminishing the thermal rate constant with Erv rise in the low exergonic area. To reveal the nonequilibrium effects on CR kinetics in Figure 5 comparison of FEG laws for thermal charge transfer and nonequilibrium CR is presented. In the region of strong exergonicity, −∆GCR > Erm , nonequilibrium kinetics is slower than the thermal whereas in the region of low exergonicity the inverse ratio of the rates is predicted. Another regularity is an increase of the dynamic solvent effect on the CR kinetics with the rise of the FEG and its decrease in the thermal charge transfer. As was mentioned above the opposite dependencies of the dynamic solvent effect on FEG for thermal electron transfer and CR are caused by



10

kCR, ps-1

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1

0.1 -0.2 -0.4 -0.6 -0.8

-1

-1.2 -1.4

∆GCR, eV Figure 5: Influence of nonequilibrium on the free energy gap law. The stochastic simulations of the thermal charge transfer kinetics (two level model) are presented by the solid lines (fast solvent) and the dashed lines (slow solvent). The stochastic simulations of CR kinetics ⊙ are pictured with signs + (the fast solvent) and (the slow solvent). Vg = 0.03 eV (red), Vg = 0.07 eV (black), τv = 0.1 ps. The rest parameters are the same as in Figure 2 caption. different initial positions of the wave packets on the CSS term. For both the thermal and nonequilibrium reactions the influence of electronic coupling variation on the rate constant increases with the rise of the FEG. This effect is slightly weaker for nonequilibrium CR. So, one can see that the manifestation of the nonequilibrium effects on the charge transfer kinetics strongly depends on the system parameters. The nonequilibrium can both speed up and slow down the charge transfer. Figure 6 shows the influence of the relaxation time of the high-frequency vibrational modes on the free energy gap law. There are two trends: (i) shortening τv leads to increase of the ET rate constant in the region −∆GCR < Erm and produces reverse and weaker effect in the Marcus inverted region; (ii) shortening τv results in a shift of the rate maximum to the area of smaller free energy gap values. The first trend directly follows from eq 25, that predicts influence of τv on the nonequilibrium CR only for terms intersections typical for the Marcus normal region but not for the Marcus inverted region. Equation 25 also predicts


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10 (1) (2)

kCR, ps-1

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(3)

1

-0.2

-0.4

-0.6

-0.8

-1

-1.2

-1.4

∆GCR, eV Figure 6: Influence of the vibrational relaxation time on the free energy gap law. (1) τv = 0.05 ⊙ ps (pictured with ), (2) τv = 0.1 ps (pictured with ×), (3) τv = 1.0 ps (pictured with +), Vg = 0.07 eV, the fast solvent. The rest parameters are the same as in Figure 2 caption. that the probability WCR increases with decreasing τv . In the Marcus inverted region CR mainly proceeds from the thermalized state of the solvent, that is, through the sinks placed in the neighborhood of the CS state minimum. But a part of systems in vibrational excited states survives and this part increases with rise of the vibrational relaxation time, τv . Since the vibrational relaxation time decreases with increasing vibrational quantum number (see eq 11) the first excited state predominantly survives. The ratio of the CR rate constants ∗ from the first excited, kCR , and the ground vibrational states, kCR , can be estimated by the

ratio of corresponding Franck-Condon factors 71 ∗ F1α nα (nα − Sα )2 kCR ∼ = kCR F0α (n−1)α nα

(28)

where α is the index of the excited vibrational mode, nα is the number of a sink in a vicinity of the minimum of the free energy curve UCSS . In the inverted region typically nα > 1 and Sα is small so that the inequality (nα − Sα )2 /nα > 1 is fulfilled. This means that the CR rates from the excited vibrational states are larger than that from the ground vibrational 19 ACS Paragon Plus Environment


10 (8)

kCR, ps-1

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(5)(6)(4)

1 (1) (3)

0.1

(2)

-0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6

∆GCR, eV Figure 7: Fitting the dependence of kCR on the free energy gap to the experimental data. ⊙ The experimental data are pictured with signs , stochastic modelling with +, and the thermal rate constant, eq 23, with solid line. The parameters of the stochastic simulations of CR rate constant in water solution are: 57 (x1 = 0.48, x2 = 0.2, x3 = 0.32, τ1 = 0.026 ps, ex τ2 = 0.126 ps, τ3 = 0.880 ps), τv = 1.0 ps, Vg = 0.03 eV, Ve = 0.025 eV, Erm = 0.001 eV, CSS Erm = 0.65 eV, Erv = 0.2 eV, ∆GGS = −2.02 eV. state. This trend is responsible for the increase of the CR rate constant with τv rise in the inverted region. The experimental data on FEG law for CR rate constant are limited by rather large values of the free energy gap. 12,31–40 To show how the model predictions are related with the available experimental data a comparison of the obtained results on ultrafast CR kinetics with the data recently reported in ref 40 are pictured in Figure 7. The experimental data were obtained for the complexes consisting of zinc(II) meso-tetrasulfonatophenylporphyrin (ZnTPPS4− /MVCN2+ ) and magnesium(II) meso-tetrasulfonatophenylporphyrin (MgTPPS4− /MV2+ ) as the electron donor and a series of viologens as the electron acceptors: methyl viologen (MV2+ ), benzyl viologen (BV2+ ), 1-methylbipyridine (MB+ ), 1,1′ -dimethyl-2,2′ -bipyridine (DM2+ ), 1,1′ ,4,4′ -tetramethyl-2,2′ -bipyridine (4,4′ -Me2 -DM2+ ), 1,1′ -bis(cyanomethyl)-4,4′ -bipyridinium (MVCN2+ ) in water solution. The numbering of the complexes in Table 2 is the same as in ref 40. These numbers are used in Figure 7 to specify the complex. 20 ACS Paragon Plus Environment

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Table 2: The experimental data of charge-transfer rate constant kCR in porphyrin/viologen complexes. 40

No 8 5 6 4 3 2 1

porphyrin/viologen complexes −∆GCR , eV kCR , ps−1 4− 2+ ZnTPPS /MVCN 1.05 4.55 4− 2+ MgTPPS /MV 1.14 1.60 4− 2+ ZnTPPS /BV 1.22 1.41 ZnTPPS4− /MV2+ 1.31 1.49 4− ′ 2+ ZnTPPS /4,4 -Me2 -DM 1.41 0.21 4− + ZnTPPS /MB 1.57 0.14 4− 2+ ZnTPPS /DM 1.59 0.57

The experimental data occupy the region of rather large values of FEG. In this region the difference between results of the stationary theory (solid line) and the stochastic simulation of nonequilibrium CR is not large and both approaches equally well fit to the experimental data that have relatively large scattering. At the moment there are systematic experimental investigations of CR rate dependence on the free energy gap only for CR into the ground electronic state. Unfortunately, there are no donor-acceptor pairs with small enough FEG to check the predictions made in the paper. On the other side, studies of the kinetics of charge transfer from the second excited state are currently intensively developed where CR in the first excited state plays paramount role. 18,41–52 In such processes the CR in the first excited state is ultrafast even when the free energy change is positive. 45,46,51,53,54 Given the importance of these processes in the operation of promising photovoltaic devices, the study of the kinetics of ultrafast recombination seems urgent experimental task. The results obtained in this paper shed some light on such CR kinetics.



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Conclusions The simulations of ultrafast CR kinetics in the framework of the multichannel stochastic model allowed us to identify differences and similarities in trends that are inherent to nonequilibrium CR reactions and the thermal charge transfer. A rather deep analogy between CR ensuing photoinduced electron transfer and that in excited donor-acceptor complexes leads to similar regularities in their kinetics. In particular, the Marcus normal region is strongly suppressed and shifted to the region of the smaller values of the FEG. The maximum of CR rate constant lies in the area of FEG values considerably smaller than that expected for the thermal reactions −∆GCR = Erm + Erv . The main conclusions on the regularities of ultrafast CR kinetics can be summarized as follows. • Slow CR, kCR ≪ 1/τRel , where τRel is an effective solvent relaxation time constant, proceeds in the thermal regime independently of the way of the initial state formation. Such reactions are expected to follow the FEG law observed for the thermal reactions. • The FEG law for ultrafast CR kinetics, kCR > 1/τRel , can strongly deviate from that observed for the thermal electron transfer. The dependence of ultrafast nonequilibrium intramolecular CR rate constant on the FEG in the neighborhood of it maximum is much weaker than that expected for the thermal ET reactions. • Manifestation of the dynamic solvent effect in the nonequilibrium CR has its peculiarities. It is large in the area of strong exergonicity and it is small in the area of weak exergonicity. This trend is opposite to that observed in the thermal reactions. • FEG laws for CR ensuing photoinduced electron transfer and CR in excited donoracceptor complexes are similar but different from that of the thermal reactions. The rate constant of ultrafast CR occurring in nonequilibrium mode is nearly monotonically decreasing function of the FEG, that is, the Marcus normal region is absent. • The CR rate constant strongly depends on the vibrational relaxation time, τv , in the 22 ACS Paragon Plus Environment

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region of weak exergonicity, −∆GCR < Erm , and weakly depends in the region of strong exergonicity. The degree of the NMR suppression strongly increases with decreasing the relaxation time of the high-frequency vibrational modes.

Acknowledgement The study was performed by a grant from the Russian Science Foundation (Grant No. 1613-10122).



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Free-Energy-Gap Law for Ultrafast Charge Recombination of Ion Pairs

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