Effect of Reactant and Product State Decay on Ultrafast Charge

Article pubs.acs.org/JPCC

Effect of Reactant and Product State Decay on Ultrafast ChargeTransfer Kinetics: Violation of the Principle of Independence of Elementary Chemical Reactions Valentina A. Mikhailova and Anatoly I. Ivanov* Volgograd State University, University Avenue 100, Volgograd 400062, Russia ABSTRACT: Fast decay of both the reactant and product states is shown to strongly increase the intrinsic electron transfer rate in donor−acceptor dyads. The decay is associated with redistribution/ relaxation of excited vibrational states that typically participate in the reaction. Although the role of the reorganization of high-frequency vibrational modes in electron-transfer dynamics is well understood and it is commonly accepted to strongly affect the ultrafast electron transfer dynamics, the influence of the relaxation of excited vibrational states on electron transfer is not accounted for in experimental data analysis. In photoinduced electron transfer, excited states of high-frequency vibrational modes are often produced by laser pump pulse so that the ultrafast charge separation, at least partly, occurs from excited vibrational states. The charge recombination accompanying the charge separation also essentially occurs from excited vibrational states of intermediates to form a final excited vibrational state. Since the decay of the reactant state and electron transfer are two elementary chemical reactions occurring in parallel, the influence of vibrational relaxation on the intrinsic electron-transfer rate constant is a clear manifestation of the violation of the fundamental principle of chemical kinetics postulating the independence of elementary chemical reactions. The mechanism of the violation is discussed in detail. The transition probability that better characterizes the efficiency of ultrafast nonequilibrium charge recombination than the rate constant is calculated. The dependencies of the electron-transfer rate constant and the transition probability on the product decay time are predicted to be identical while on the reactant decay time to be opposite.

■

INTRODUCTION Ultrafast transfer of charge (electron, proton, hole) is essential in fundamental science and diverse technological applications.1−10 Currently, there are emerging applications in the areas of electrocatalysis, solar energy conversion, and molecular electronics.11−16 Since the ultrafast electron transfer (ET) typically proceeds on a time scale comparable with that of the relaxation of solvent and intramolecular vibrations17−21 the nuclear nonequilibrium created by an excitation laser pulse or chemical transformation may be of paramount importance in such reactions.22−28 The regularities governing the kinetics of electron transfer occurring from nonthermalized solvent29−31 and nonequilibrium state of nuclear subsystem of the reactants32−34 can drastically differ from that of the thermal reactions. As a result, interest is growing in the study of such processes.9,10 The dependence of electron-transfer rate constant on the excitation pulse carrier frequency observed in ultrafast charge recombination of excited donor−acceptor complexes (DAC) in polar solvents is a clear manifestation of the solvent nonequilibrium created by the pump pulse.22−24 It has recently been shown that the dynamic solvent effect observed in the region of strong exergonicity for ultrafast charge recombination in excited DACs35 is even more convincing proof that the reaction proceeds in nonequilibrium regime.31 The effects of the excitation pulse carrier frequency and the solvent viscosity © 2017 American Chemical Society

on ultrafast charge recombination in DACs were rather well described in the framework of the stochastic multichannel point-transition model.22,31 The intramolecular high-frequency mode excitation by a pumping pulse also can affect the ultrafast ET kinetics. Recently, a great deal of experimental evidence on the influence of high-frequency intramolecular vibration excitation on the ET dynamics has been obtained.25−27,36−46 Excitation of highfrequency vibrational modes can alter both the forward and backward ET rates.46 Theoretical simulations of the effect of intramolecular highfrequency vibrational mode excitation on photoinduced ET also have been carried out.32−34,47 The theoretical results are, in general, consistent with the experimental data. Depending on the parameters of DACs, the excitation of high-frequency vibrational modes can either increase or decrease the photoinduced ET rate by several tens of percent.34,46 Thus, the experimental data indicate that ultrafast ET can proceed from excited vibrational states, and the theoretical simulations confirm this conclusion. The excited vibrational states decay with time constant τv due to vibrational relaxation. Ultrafast ET from the excited vibrational state and its decay occur in parallel; Received: June 21, 2017 Revised: August 31, 2017 Published: September 6, 2017 20629

DOI: 10.1021/acs.jpcc.7b06106 J. Phys. Chem. C 2017, 121, 20629−20639

Article

The Journal of Physical Chemistry C

devices based on dye-sensitized solar cells.9 Since the charge recombination in many applications is undesirable process, it is important to know how to suppress it. To implement this one should know detailed microscopic mechanism of ultrafast charge recombination and dependencies of its effective rate constant on controlling parameters. The main aims of this paper are (i) to derive explicit expressions for the intrinsic rate constant of ET and the probability of nonthermal electronic transition accounting for the decay of the reactant and product states, (ii) to determine the extent of the violation of the principle of elementary reactions independence in the ultrafast ET, and (iii) to clarify the mechanism of the influence of the decay time scale on the probability and the rate constant of ultrafast ET.

in other words, there is a competition between them. In the simplest case of irreversible ET its exponential kinetics can be described by the set of equations33 ⎛ dN11 1⎞ = −⎜k1 + ⎟N11 τv ⎠ dt ⎝

(1)

⎛ dN10 1⎞ = −k 0N10 + ⎜WNk1 + ⎟N11 τv ⎠ dt ⎝

(2)

where N11 and N10 are the populations of the excited and ground vibrational states of the reactants, k1 and k0 are the rate constants of the ET from these states, correspondingly, and WN is the probability of the nonthermal back ET (see Figure 1).

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THEORY AND COMPUTATIONAL DETAILS Reorganization of intramolecular high-frequency vibrational modes strongly affects the ET kinetics.10,29 The reorganization often results in population of excited states of vibrational modes of products. Nonequilibrium ultrafast charge recombination also proceeds to form the products in excited vibrational states.31 Typically, the decay of excited vibrational states is extremely fast due to intramolecular vibrational redistribution and relaxation.50 Influence of the product state decay on the probability of nonequilibrium ET was investigated in refs 51 and 52. The ET probability was shown to strongly increase with decreasing lifetime of excited vibrational states. Photoinduced ET can be so fast that it can effectively compete with vibrational relaxation.25,46 This means that ET can occur from excited vibrational states when the pump pulse populates them. Experimental evidence of the influence of highfrequency intramolecular vibration excitation on the ET dynamics requires establishing the amount of the effect of the reactant state decay on the ET kinetics. In this section, we study influence of the reactant state decay on ET kinetics using a simple two-level model for donor−acceptor dyads DA. The model involves only two states of a dyad: a neutral state, DA, and a charge-separated state, D+A−, and allows electron transfer between them to be described. Simplicity of the model allows analytical expressions for the ET rate constants to be obtained, as well as the probability of nonequilibrium ET and clarification of the physical mechanism of the influence. The model is applicable for a quantitative description of electron transfer in different donor−acceptor systems, for example, photoinduced electron transfer in donor−acceptor dyads and pairs and charge recombination in excited DACs. In what follows, the influence of the relaxation of a quantum vibrational mode with the frequency ℏΩ ≫ kBT on ET rate is studied. It is supposed that the Huang−Rhys factor of the vibrational mode at the electron transfer stage is not equal to zero. This means that electron transfer in a series of states with a different number of vibrational quanta is allowed. In the general case, a number of the transitions proceed in parallel; that is, the reaction is multichannel. The condition ℏΩ ≫ kBT was shown (see Appendix in ref 53) to guarantee the independence of the channels. In this case, the rate constants for individual channels provide more detail information on electron transfer kinetics than the total rate. Moreover, the reactions in the Marcus normal region proceed mainly to form the product in the ground vibrational state due to considerably larger activation barrier for the channels with excited vibrations of the products. In this case, the reaction is essentially a single

Figure 1. Free energy curves of the ground, S10, and first excited vibrational, S11, sublevels of the locally excited electronic state and the charge separated state (CSS). Photoinduced ET and nonthermal charge recombination are pictured with green and black curved arrows, correspondingly. The vertical blue and red arrows stand for the vibrational (VR) and solvent relaxation, respectively. The blue bell visualizes the initial distribution of the particles in the reactant state.

One of the fundamental principles of the formal kinetics is the assumption of the independent course of elementary reactions. According to this postulate introduced by Ostwald, the rate constant of an elementary chemical reaction does not depend on whether other elementary reactions occur in the given system simultaneously.48 In this study, the ET rate constant k1 was shown to depend on the vibrational relaxation time τv that clearly demonstrates the violation of this postulate. The violation of Ostwald’s postulate is expected when an ultrafast reaction disturbs the thermal distribution along the reaction coordinate. Obviously, this disturbance affects the rate constant of the second reaction occurring in parallel; hence, the reactions are not independent. Considered here, the violation of the Ostwald’s postulate has different nature. Indeed, the vibrational relaxation does not disturb the thermal distribution along the reaction coordinate; nevertheless, it can considerably increase the ET rate constant. Apparently, the latter extends the region where the violation of Ostwald’s postulate occurs. The effect of the reactant state decay is operative not only in photoinduced ET where intramolecular vibrations are excited by a pump pulse. The effect can be of even more importance in charge recombination following direct photoinduced ET. Such a charge recombination also often proceeds in ultrafast regime and the excitation of intramolecular high-frequency vibrations are produced by the preceding charge separation in complex. Typically, charge recombination results in the loss of the energy or selectivity of a photoreaction. For example, it significantly suppresses the efficiency of photocatalysis49 and photovoltaic 20630


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Here, Qmin = 2Erm, Q* = Erm + ΔGET + (np − nr)ℏΩ, Erm is the solvent reorganization energy, and ΔGET is the ET free energy change. For a description of the quantum vibrational mode, the Bixon−Jortner model is used.61 The quantities ρr(Q,t) and ρp(Q,t) are the distribution functions of the particles in the reactant and product electronic states with nr and np vibrational quanta of high-frequency vibrational mode, correspondingly. The last term in eq 3 describes transitions from the reactant electronic state with nr vibrational quanta to the reactant state with nr − 1 vibrational quanta, that is, vibrational relaxation. The sense of the last term in eq 4 is analogical. We are interested here in the kinetics of the transitions between the reactant and product states with nr and np vibrational quanta; therefore, we do not consider the fate of the states with nr − 1 and np − 1 quanta. The concept of parabolic free energy curves constructed in the space of solvent polarization coordinate lies in the base of the model considered.62 Diffusion motion of the particles along these curves reflects the reorganization of a polar solvent and is fully characterized by the solvent relaxation function X(t) = ⟨Q(t)Q(0)⟩/⟨Q(0)Q(0)⟩, where the brackets stand for an ensemble average. The parameters of the relaxation function of a specific solvent can be obtained from experimental data on the dynamic Stokes shift.19,63−67 When solvent relaxation is characterized by a single Debye relaxation time, the autocorrelation function has the form X(t) = e−t/τL and the diffusion on the free energy curves Uα (α = r, p) is described by the Smoluchowski operators L̂ α:

channel and the calculated rate is close to the total rate constant measured in experiments. The kinetics of ET between individual vibrational sublevels can be described in the framework of the stochastic approach54 generalized to account for the decay of both the reactant and the product states. A set of two coupled differential equations22,23,28,51,52,54−60 ∂ρr ∂t ∂ρp ∂t

= L̂r ρr − (2πVel2 /ℏ)δ(Q − Q *)(ρr − ρp ) −

= L̂pρp + (2πVel2 /ℏ)δ(Q − Q *)(ρr − ρp ) −

ρr τvr

(3)

ρp τvp

(4)

for the probability distribution functions along the reaction coordinate, Q, for the reactant state, ρr(Q,t), and the product state, ρp(Q,t) is solved analytically by Green’s function method. Here, ℏ is the Planck constant, and δ(Q) is the Dirac delta function. The transition between the excited vibrational sublevels of the reactant (nr) and the product (np) states (see

∂U ∂ 2Erm ⎛ ∂ 2Uα ∂2 ⎞ ⎜ 2 + α + kBT 2 ⎟ Lα̂ = τL ⎝ ∂Q ∂Q ∂Q ∂Q ⎠ Figure 2. Free energy curves of the ground (solid lines) and the excited (dashed lines) vibrational sublevels of the reactant, Ur, and product, Up, states as functions of the classical reaction coordinate Q. The black bell visualizes the initial distribution of the particles in the reactant state that is associated with a vibrational sublevel of a locally excited state of a donor−acceptor dyad. The ET is pictured by a red arrow. The vertical red and blue arrows stand for the decay of the reactant and product states.

Here, the relaxation time τL is usually related to the time of longitudinal dielectric relaxation, kB is the Boltzmann constant, and T is the temperature. The set of eqs 3 and 4 is solved with the initial conditions ρr (Q , t = 0) = φ(Q ),

Vel =

V el(0)e−S /2

nr ! n p!

∑ α=0

ρp (Q , t = 0) = 0

2πVel2 ℏ

ρr (Q , t ) = φ(Q , t ) −

( −1)nr − α S(nr + np − 2α)/2 α ! (n r − α ) ! (n p − α )!

∫0

t

dt ′

[ρr (Q *, t ) − ρp (Q *, t )]Gr(Q , t − t ′|Q *) (9)

(5)

where S = Ervib/ℏΩ and Ervib are the Huang−Rhys factor and the reorganization energy of high-frequency vibrational mode with the frequency Ω, respectively. These excited vibrational states of the reactant and product rapidly decay with the rate constants 1/τvα (α = r, p) due to vibrational redistribution/ relaxation.50 In eqs 3 and 4, the coordinate Q* corresponds to the crossing point of the free energy curves Ur(Q*) = Up(Q*) where Q2 Ur = + nrℏΩ, 4Erm

Up =

(Q − Q min)2 4Erm

(8)

The form of the function φ(Q) is determined by the conditions of the preparation of the initial state. Using Green’s function techniques, eqs 3 and 4 can be rewritten as

Figure 2) is induced by the electronic coupling Vel, which accounts for the Franck−Condon factor and is equal to29 min(nr , n p)

(7)

ρp (Q , t ) =

2πVel2 ℏ

∫0

t

dt ′[ρr (Q *, t ) − ρp (Q *, t )]Gp

(Q , t − t ′|Q *)

(10)

where ∞

φ (Q , t ) =

∫−∞ dQ ′Gr(Q , t|Q ′)φ(Q ′)

Gr (Q , t |Q ′) =

+ npℏΩ + ΔG ET (6)

(11)

⎧ (Q − Q ′e−t / τL)2 ⎫ ⎬ exp⎨− 2σ 2(t ) ⎩ ⎭ 2πσ 2(t ) e−t / τvr

(12) 20631


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The Journal of Physical Chemistry C Gp(Q , t |Q ′) =

e−t / τvp

Next, we apply the Laplace transform to eqs 9 and 10

2

2πσ (t )

f ̃ (s ) =

⎧ [Q − Q min − (Q ′ − Q min)e−t / τL]2 ⎫ ⎪ ⎪ ⎬ − exp⎨ 2 ⎪ ⎪ σ 2 ( t ) ⎭ ⎩ (13)

Pp̃ (s) =

Pα(t ) =

(2πVel2 /ℏ)φ(̃ Q *, s) (s + 1/τvp)[1 + (2πVel2 /ℏ)(Gr̃ (Q * , s|Q *) + G̃p(Q * , s|Q *))]

(16)

⎧ Q2 ⎫ ⎬ φth(Q ) = (4πErmkBT )−1/2 exp⎨− ⎩ 4ErmkBT ⎭

fvα =

G̃p(Q *, s|Q *) ≃

1 |A r |fvr

φth(Q * − Q min) s + 1/τvp

1 |A p|fvp

(19)

where Pr̃(s) =

1+

Ap =

|Q * − Q min|

8ErmkBT Aα2 τLτvα

τL

,

α = r, p

k rp(s + 1/τvp) + k pr(s + 1/τvr) + [1 − Φ(Q *, s)](s + 1/τvr)(s + 1/τvp) (s + 1/τvr){(s + 1/τvr)(s + 1/τvp) + k rp(s + 1/τvp) + k pr(s + 1/τvr)}

Pp̃ (s) =

(21)

Φ(Q *, s)(s + 1/τvp) (s + 1/τvr)(s + 1/τvp) + k rp(s + 1/τvp) + k pr(s + 1/τvr)

(22)

Here, g is the reaction nonadiabaticity parameter that depends not only on the electronic coupling and the term slopes at the intersection point, Q*, but also on the factors f vr and f vp, and

where Φ(Q *, s) =

g=

Q* , τL

Here Ar and Ap are the average (drift) velocities of the particles in the vicinity of the term intersection point, Q*, for the reactant and product states, and f vα is the factor that accounts for the escape of particles from the reaction zone (region of term intersection) due to the decay of the reactants (α = r) and products (α = p). Insertion of the asymptotic expansions 18 and 19 in eqs 15 and 16 leads to the final results

(18)

+

(20)

is the thermal distribution in the reactant state

In this case, it suffices to know only the long-time asymptotic of the Green’s functions.54 For this purpose, we expand the Laplace transforms of the Green’s functions in powers of s ignoring all the terms with positive powers of s52,68 s + 1/τvr

(14)

(15)

Ar =

+

α = r, p

(s + 1/τvr){1 + (2πVel2 /ℏ)[Gr̃ (Q *, s|Q *) + G̃p(Q *, s|Q *)]}

(Q * − Q min)2 /4Erm ≫ kBT

φth(Q *)

∫ ρα (Q , t ) dQ ,

1 + (2πVel2 /ℏ)[Gr̃ (Q *, s|Q *) + G̃p(Q *, s|Q *) − φ(̃ Q *, s)]

(17)

Gr̃ (Q *, s|Q *) ≃

e−st f (t ) dt

to obtain

Equations 15 and 16 describe the ET dynamics in a system with decaying products and reactants, starting with an arbitrary initial state of the nuclear subsystem (equilibrium and nonequilibrium). To understand the structure of the solutions obtained, let us consider ET with high activation energy Q *2 /4Erm ≫ kBT ,

∞

and pass to the reactant and product populations

are the Green functions for eqs 3 and 4 without δ-sinks. Here, σ2(t) = 2ErmkBT(1 − e−2t/τL). Pr̃(s) =

∫0

2πVel2 φ(̃ Q *, s) ℏ(1 + g )

⎞ ⎛ 2πVel2 ⎜ 1 1 ⎟ + ℏ ⎜⎝ |A r |fvr |A p|fvp ⎟⎠

(23) 20632

k rp =

2πVel2 φ (Q *) ℏ(1 + g ) th

(24)

k pr =

2πVel2 φ (Q * − Q min) ℏ(1 + g ) th

(25)


Article

The Journal of Physical Chemistry C are the intrinsic thermal rate constants for the forward and backward ET also accounting for the decay of the reactant and product states. Equations 21, 22, 24, and 25 describe the ET dynamics of the system with an arbitrary initial state of the nuclear subsystem. They are a modification of well-known Zusman equations for the reaction rate constants that bridged between the nonadiabatic, g ≪ 1, and the solvent controlled, g ≫ 1, regimes.54 In the nonadiabatic limit the dependencies of the intrinsic thermal rate constants 24 and 25 on the decay times of both reactants and products disappear. In the opposite limit, g ≫ 1, the influence of the reactant and product state decay on the ET kinetics can be significant. The most general conclusion ensuing eqs 24, and 25 is the decay of both the reactant and product states increases the ET rate constant. To clarify the meaning of the reaction rate constants given by eq 24, we note that in the general case the ET kinetics is not exponential. This is obvious for nonequilibrium initial conditions. For the initial equilibrium distribution the nonexponentiality is caused by a distortion of the distribution by fast reaction itself. So, strictly speaking, the rate constant of fast reactions does not exist. In such a case an effective rate constant is usually exploited. The effective rate constant is typically determined by an equation of the kind of eq 31. The long-time asymptotic of the Green’s functions leads exactly to such an effective rate constant. This can be easily seen from eqs 21 or 27 in the case of irreversible reaction (kpr= 0). In this case, one gets P̃ r(s = 0) = ∫ ∞ 0 Pr(t) dt = 1/(kpr + 1/τvr) that uncovers the physical meaning of the rate krp that is in full accord with eq 31. Obviously, the effective rate constant accounts for the ET dynamics at the early stage when the both quantum and classical modes are in nonequilibrium states. The mechanism of the influence of the product state decay in the solvent-controlled regime on the ET rate constant is fairly transparent. A particle that reaches the transition state can many times intersects the term-crossing point before leaving it due to diffusional motion, so that there are forward and backward reaction fluxes. The product state decay suppresses the backward flux that results in an increase of the forward rate constant. It should be stressed that we speak here about microscopic reversibility. The reverse flux of the particles after their thermalization is not considered. Thus, the difference of the forward and backward fluxes is equal to the forward reaction rate. The mechanism of the reactant state decay impact on the rate constant is more complex and will be considered below. The effect of τvr and τvp on ET rate is a clear manifestation of the violation of the elementary chemical reaction independence principle. To quantitatively determine the scale of the effect of the reactant and product state decay on the ET rate constant and the transition probability, the ET kinetics from thermal and nonequilibrium states of the solvent will be considered in the next section.

In this case, eqs 21 and 22 are rewritten as Pr̃(s) =

Pp̃ (s) =

k rp (s + 1/τvr)(s + 1/τvp) + k rp(s + 1/τvp) + k pr(s + 1/τvr)

(28)

The originals of the Laplace transforms eqs 27 and 28 satisfy the following kinetic equations for the reactant and product populations dPr = −(k rp + 1/τvr)Pr + k prPp dt

dPp dt

= k rpPr − (k pr + 1/τvp)Pp

(29)

(30)

with initial conditions Pr(0) = 1, Pp(0) = 0. These equations completely clarify the physical meaning of the rate constants 24 and 25. The dependencies of the rate constant for the forward reaction on the free energy change and on the reactant state decay time are presented in Figures 3 and 4. Hereinafter the

Figure 3. Free energy change dependencies of the thermal rate constant for the forward ET reaction, krp, eq 24, are shown by the solid lines. Numerical data are pictured by separate dots. Parameters used: Vel = 0.02 eV, τL = 5 ps, Erm = 0.6 eV, τvp = 100 ps, τvr = 100 ps (black dots and solid lines 1), τvr = 1 ps (red dots and solid lines 2), and τvr = 0.15 ps (blue dots and solid lines 3).

solvent reorganization energy, the electronic coupling, and the temperature are fixed: Erm = 0.6 eV, Vel = 0.02 eV, kBT = 0.025 eV. Variable parameters of the stochastic model in the calculations are the free energy gap, −ΔGET, the solvent relaxation time, τL, and the decay times of the reactants and products, τvα (α = r, p). For the accepted values of the parameters ET proceeds in the solvent controlled regime. For example, for ΔGET = −0.3 eV the value of g alters with the change of the variable parameters, τvp and τvr, in the interval from 82 to 30. Figure 3 shows that there exists a wide region of the exergonicity of the ET reaction, in which the effect of the reactant state decay on the rate constant is strong. What is especially important is that the effect is large in the neighborhood of the barrierless region where the ET rate is maximum. In this region, the ET from an excited vibrational state can effectively compete with vibrational relaxation. As a result, the quantum yield of the products from the excited vibrational state can be large. In areas where the free energy barrier for a forward reaction is small, the inequalities 17 are not satisfied, and hence, eqs 24

RESULTS AND DISCUSSION Electron Transfer from Thermal State of Solvent. When ET in a donor−acceptor systems proceeds from the thermalized state of the solvent, the initial distribution function φ(Q) = φth(Q). The Laplace transformation of φ(Q,t) = φth(Q)exp(−t/τvr) is φth(Q *) s + 1/τvr

(s + 1/τvr)(s + 1/τvp) + k rp(s + 1/τvp) + k pr(s + 1/τvr)

(27)

■

φ(̃ Q *, s) =

k pr + (s + 1/τvp)

(26) 20633


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ET practically irreversibly proceeds. In this case, the time dependence of the reactant population is transformed to the form Pr(t) = exp(−krpt − t/τvr). Since the decay of the reactant state and ET, in fact, are two elementary chemical reactions occurring in parallel, the influence of τvr on ET rate is a clear manifestation of the violation of the elementary chemical reaction independence principle.48 The mechanism of violation of this principle is discussed in detail in the next subsection. Mechanism of Violation of the Principle of Independence of Chemical Reactions in the Case of ET and Vibrational Relaxation Occurring in Parallel. To clarify the mechanism of violation of the principle of independence of chemical reactions, we confine ourselves to a simpler reaction where the ET proceeds in irreversible mode. In this case, we obtain from eq 27 Pr̃(s) =

1 s + 1/τvr + k rpir

(32)

where k rpir = Figure 4. Dependencies of the ET rate constant, krp, on the time of the reactant state decay, τvr. Numerical data for the krp are pictured by the symbol line. Parameters used: Vel = 0.02 eV, Erm = 0.6 eV, τL = 0.5 ps (frame a), τL = 5 ps (frame b). The values of the free energy gap, −ΔGET in eV, are shown near the lines.

∫0

∞

Pr(t ) dt )−1 − 1/τvr

ℏ(1 + g ir )

φth(Q *)

(33)

is the thermal ET rate constant g ir =

2πVel2 ℏA r fvr

The reverse Laplace transformation of eq 32 determines the time evolution of the reactant population as follows

and 25 are not applicable. In this case, the system of eqs 3 and 4 with the initial condition 20 is solved numerically using the Brownian simulation method.23,69,70 In the numerical solution, the effective rate constant of ET is determined as ̃ =( k rp

2πVel2

Pr(t ) = exp[− (1/τvr + k rpir)t ]

(34)

kirrp

This equation shows that is the rate constant of the ET. Equation 33 can be rewritten in the following form

(31)

⎛ 1 1 1 ⎞ + ⎟ ir = ⎜ k SC ⎠ k rp ⎝ kNA

The results of numerical simulations of the ET kinetics are pictured in Figure 3 by dots. Analytical expressions for the rate constants are in a good agreement with the numerical results in the areas of their applicability. Figure 3 demonstrates the conclusion ensuing from eqs 24 and 25: for ET reactions with a low barrier of the forward stage, −Erm − 8kBT < ΔGET < −Erm + 8kBT, the rate constant of this stage, krp, is affected only by the decay of the reactant. The decay of the product state has a negligible effect, since the backward reaction flux is small due to the large slope of the product term at the point of electronic transitions and its further suppression has little effect. The acceleration of the ET with decreasing time constant of the reactant state decay, τvr, is demonstrated in Figure 4. For barrierless ET (black symbol lines) in a slow solvent, such as valeronitrile, τL = 5 ps (frame b), the effect is more noticeable than that in a fast solvent, such as acetonitrile, τL = 0.5 ps (frame a). For the slow solvent, the rate constant, krp, increases by a factor of 5 with decreasing time scale τvr to 0.05 ps, and in the fast solvent only by 1.7 times. For the ET with a low free energy barrier, −ΔGET = 0.8 and 0.4 eV, the effect is more noticeable (an increase by a factor of 3) in fast solvents (blue and red lines, frame a) than in slow one (blue and red lines, frame b, an increase by a factor of 2). Growing the free energy barrier for ET weakens the influence of the solvent relaxation time scale, τL, on the dependence krp(τvr). It should be emphasized that for the exergonicity region −1.1 eV < ΔGET < −0.2 eV a relation krp ≫ kpr is satisfied and the

(35)

where kNA =

2πVel2 φ (Q *) ℏ th

k SC = A r fvr φth(Q *)

(36) (37)

In the extreme of fast diffusion (small τL) the inequality kSC ≫ kNA is held and kirrp ≃ kNA, where kNA coincides with the Golden Rule results. This is the nonadiabatic limit, and the reaction rate is limited by the electronic transition between the reactant and product states. This limit corresponds to the kinetic regime. In the opposite extreme, kSC ≪ kNA, we obtain kirrp ≃ kSC. This is the solvent-controlled regime, and the reaction is determined by the diffusional delivery of the particles to the reaction zone (vicinity of the term crossing). Naturally, in the nonadiabatic limit the vibrational relaxation does not influence on the ET rate constant and kNA does not depend on magnitude of τvr. On the contrary, in the solvent controlled regime ET rate constant strongly depends on τvr. To appreciate the physics underlaying this dependence we start with calculation of kSC at qualitative level following the ideas reported in ref 71. The transition state region (TSR) plays a central role in the approach. It is determined as an interval of reaction coordinate values in vicinity of the sink point, Q* = ΔGET + Erm, where the Ur(Q*) − Ur(Q) < kBT. In the linear approximation of the 20634


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operative only if the inequality lf > lv is fulfilled, where lf is the free mean path length along the reaction coordinate. Nonthermal ET Reaction. Ultrafast charge recombination in excited donor−acceptor complexes and ensuing photoinduced ET proceeds in nonthermal mode.29,31 Pumping pulse transfers vertically a part of population that can be visualized as appearance of a wave packet above the term crossing. Photoinduced ET occurring in the vicinity of the term crossings also populate the term of the charge transferred state above the crossing point of the charge-transferred and ground states where charge recombination can proceed (see Figure 1). The motion of such a wave packet toward the term bottom reflecting the relaxation of the solvent to the new equilibrium position results in its passing through the term crossing where the electronic transitions occur. The effectiveness of such transitions is described by the probability rather than by the rate. To calculate the probability of nonthermal transition, eqs 3 and 4 should be solved with the initial condition ρr(Q, t = 0) = δ(Q − Q0) where Ur(Q0) exceeds the free energy at the term crossing point Ur(Q*). The solutions 21 and 22 are recast as

terms in the TSR one obtains FlTS = kBT, where F = (ΔGET + Erm)/2Erm is the term slope at the point of the sink. The length of the TSR is 2kBTErm ΔG ET + Erm

l TS =

(38)

All of the particles occupying the TSR arrive in the sink without additional activation in time ΔtTS Δt TS =

2 l TS kBTErmτL = 2D (ΔG ET + Erm)2

(39)

where D=

2kBTErm τL

(40)

is the diffusion coefficient along the reaction coordinate Q as it follows from eq 7. The number of the particles occupying TSR is Δn TS = l TSφth(Q *)

(41)

and the CT rate constant in the solvent controlled regime is k SC =

Δn TS ΔG ET + Erm = φth(Q *) 2Δt TS τL

Pr̃(s) =

Pp̃ (s) =

Δn v = 2τvr

WN =

8Δt TS τvr

2πVel2 ℏ|A r |fvr (1 + g )

(48)

is the probability of the nonthermal electronic transitions (see Figure 1) and g is given by eq 23. It is not difficult to show that the explicit expression 48 is transformed into the well-known expression in the limits τvr → ∞ and τvp → ∞.51,52,72 In the limit of weak electronic coupling, g ≪ 1

(44)

WN =

where Δnv is the number of the particles in the interval lv. Accounting for the inequality τvr ≪ ΔtTS, the approximate equality can be obtained 1/2 ⎡ 8Δt TS ⎤ fvr = ⎢1 + ⎥ ≃ τvr ⎦ ⎣


(47)

(43)

4ErmkBT φth(Q *) τvrτL

k rp + (s + 1/τvr)WN

where

and the rate constant can be estimated as follows kv =


(46) (42)

The factor 2 in the denominator reflects the fact that only a half of the particles in the TSR are moving toward the sink. Equation 42 coincides with eq 37 for f vr = 1. To account for the reactant-state decay we notice that the decay limits the time of the reaction to the value τvr. This means that only those particles can participate in ET which arrives the sink during the time interval τvr. When the inequality τvr ≪ ΔtTS is held such particles occupy the interval of the reaction coordinate l v2 = 2Dτvr

k pr + (s + 1/τvp)(1 − WN)

2πVel2 ℏ|A r |fvr

(49)

the probability does not depend on the decay rate of the product state but still depends on the lifetime of the reactant state. In the opposite limit, g ≫ 1, the probability is independent of the electronic coupling

(45)

With this approximation eq 44 differs from exact eq 37 only by a factor of √2. Equation 45 points out that the dimensionless parameter determining the effect of the decay of the reactant state on ET rate constant is the ratio of the lifetime of the particles in the transition state and the decay time of the reactant state without ET. Thus, decreasing the time of a reactant state decay increases the ET rate constant. Such a behavior of the rate constant is a direct consequence of the diffusion motion law ΔQ2 = 2DΔt. If the particles ballistically move through the regions lTS and lv the motion law is ΔQ = vΔt where v is the average thermal velocity. The calculations show that for the ballistic motion on the time interval τvr the decay of the reactant state does not affect the ET rate constant. So, we can conclude that the effect of the reactant state decay is

WN =

|A p|fvp |A r |fvr + |A p|fvp

(50)

Comparing eqs 49 and 50 shows that the effect of the reactant state decay is most pronounced in the limit of weak electronic coupling while the effect of the product state decay is maximum in the opposite limit. The expression has a very transparent interpretation. The quantities Ar and Ap are the average velocities of the particles issuing from the reaction point along the reactant and the product terms, respectively.72 The vibrational relaxation factors, f vp and f vr, account for the disappearance of the particles due to the decay of corresponding state so that the quantities |Ap|f vp and |Ar|f vr are the total fluxes of the particles from the intersection point of 20635


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The Journal of Physical Chemistry C the initial and final terms to the product and reactant state, respectively. This makes clear the mechanism of the influence of the reactant and product state decay on the probability of the nonthermal transitions. In the solvent controlled limit a particle passes a term crossing point many times before it leaves the transition state, that is, there are direct and back fluxes between the product and reactant states. The decay of the product state decreasing the back flux increases the transition probability whereas the decay of the reactant state decreases the forward flux that results in decreasing the probability WN. The dependencies of the nonthermal ET probability, WN, eq 48, on the free energy change, ΔGET, for a few values of the solvent relaxation time scale, the product and reactant life times are pictured in Figure 5. The probability WN is not determined

product state decays increase with increasing the solvent relaxation time (compare frames a and b). The dependencies of the nonthermal ET probability, WN, on the time scale of the reactant state decay, τvr, for a few values of ΔGET in fast and slow solvents are illustrated in Figure 6. All

Figure 6. Dependencies of the nonthermal ET probability, WN, on the time constant of the reactant state decay, τvr. Parameters used: Vel = 0.02 eV, Erm = 0.6 eV, τL = 1 ps (frame a), τL = 5 ps (frame b), τvp = 100 ps (red lines), 0.15 ps (blue lines). The values of the free energy gap, − ΔGET in eV, are shown near the lines.

graphs show monotonic increase of ET probability with increasing the reactant decay time. In the region of large values of τvr the dependence approaches to a plateau. Naturally, in a slower solvent approaching to the plateau occurs in the area of larger values of τvr. The effect of the reactant state decay is rather large for all selected values of the free energy gap. Increasing the value of τvr can result in more than 3-fold increase of the ET probability. The ET can be described in terms of two different quantities, the rate constant and the transition probability. Intuitively, we consider that they should be proportional to each other; hence, they are expected to have similar dependencies on the parameters. Indeed, the dependencies of WN and krp on the product decay time are identical (compare eqs 33 and 48). However, these quantities demonstrate opposite dependencies on the reactant decay time. Thus, there is an important difference between WN and krp that must be accounted for when analyzing the thermal and nonequilibrium reactions.

Figure 5. Free energy change dependencies of the nonthermal ET probability, WN. Parameters used: Vel = 0.02 eV, Erm = 0.6 eV, τL = 1 ps (frame a), τL = 5 ps (frame b), τvr = 100 ps (black lines, 1), 1 ps (red lines, 2), 0.15 ps (blue lines, 3). The solid and dashed lines correspond to the values of the lifetime of the product state τvp = 100 and 0.15 ps, respectively.

in a neighborhood of the point ΔGET = −Erm (barrierless region); therefore, it is not shown in the vicinity of ΔGET = −0.6 eV. Figure 5 demonstrates the most general conclusion ensuing from eq 48: the decay of the product state increases the probability of the nonthermal electronic transition whereas the decay of the reactant state decreases it. It should be stressed that the decay considered here is associated with intramolecular vibrational redistribution/relaxation and it does not alter the number of the particles in the electronic reactant and product states. The maximum of the effect of the reactant state decay on the ET probability is achieved in the neighborhood of the barrierless region ΔGET ≃ − Erm being rather weak in both regions of small and large free energy change, ΔGET. The effect of the product state decay on the ET probability monotonically increases with increasing the free energy change, ΔGET. Such behavior of the effects reflects the alteration in the slopes of the reactant and the product terms at the point of their intersection with the variation of ΔGET. Both effects of the reactant and

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CONCLUSIONS The study performed has shown that the decay of the reactant and product states strongly affects the rate constant and the probability of electron transfer. To estimate the scale of the effect the typical values of vibrational relaxation time constant is exploited. The time constants are usually associated with transitions between the first excited and the ground vibrational states, 1 → 0. Electron transfer proceeding in the Marcus inverted region and nonequilibrium charge recombination 20636


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results in population of higher excited vibrational states. Quantitative information on the lifetime of higher excited states is practically absent. However, accounting for fast increase of the density of vibrational states with increasing the energy in polyatomic molecules, one can expect that the lifetime of higher excited states due to n → n − 1 (n > 1) transitions is much shorter than that for 1 → 0. As a result, in such processes the influence of vibrational redistribution/ relaxation on ET dynamics can be considerably lager than that obtained in this article. The effect of vibrational relaxation on ET dynamics is most pronounced in thermal photoinduced ET proceeding in the neighborhood of the barrierless region. It is in this region that the electron transfer can be ultrafast and can effectively compete with vibrational relaxation and, consequently, the effects under consideration play an important role. The effect is even more important in ultrafast charge recombination proceeding in nonthermal regime. Although the decay of the reactants violates the principle of elementary reaction independence, the forward and backward reaction rate constants eqs 24 and 25 satisfy the equation k rp k pr

⎧ ΔG ET ⎫ ⎬ = exp⎨− ⎩ kBT ⎭

(51)

that is nothing but the condition of detailed balance for nondecaying reactants and products. The model considered in this work supposes that the electron transfer can be described within a two-level approximation. On the other hand, the transitions between excited vibrational sublevels of the reactant and product states are studied. The sublevels form a manifold of the reactant and product states, divided by free energy gaps, equal to the highfrequency vibrational quantum, ℏΩ ≫ kBT.3 Since ET can occur at each intersection of the vibrational sublevels, the problem with a series of the term crossings should be solved. Although the transitions can affect one another, for highfrequency vibrational modes the inequality ℏΩ ≫ kBT is fulfilled by definition, which guarantees independence of the transitions.53 Thus, the two-level approximation allows us to find the probability of ET between vibrational sublevels.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Anatoly I. Ivanov: 0000-0002-4420-5863 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The study was performed with a grant from the Russian Science Foundation (Grant No. 16-13-10122).

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REFERENCES

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Effect of Reactant and Product State Decay on Ultrafast Charge

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