Effect of Excitation Pulse Carrier Frequency on Ultrafast Photoinduced

Jan 27, 2015 - ABSTRACT: Influence of excitation pulse carrier frequency on photoinduced ultrafast intramolecular charge transfer kinetics is investig...
0 downloads 0 Views 635KB Size
Download PDF
Article pubs.acs.org/JPCC

Effect of Excitation Pulse Carrier Frequency on Ultrafast Photoinduced Charge Transfer Kinetics: Effect of Intramolecular High Frequency Vibrational Mode Excitation Vadim Yu. Barykov, Vladimir N. Ionkin, and Anatoly I. Ivanov* Volgograd State University, University Avenue 100, Volgograd 400062, Russia ABSTRACT: Influence of excitation pulse carrier frequency on photoinduced ultrafast intramolecular charge transfer kinetics is investigated in the framework of a multichannel stochastic point transition model involving an excited state formation. It is supposed that an intramolecular high frequency vibrational mode being active at the excitation stage also accepts the energy at the stage of photoinduced charge transfer. A strong dependence of the photoinduced charge transfer rate constant on excitation pulse carrier frequency is uncovered. Since this dependence is associated with charge separation from different excited states of an intramolecular high frequency vibrational mode, it is named the vibrational spectral effect. The simulations show that the effect may be both positive (the photoinduced charge transfer rate constant increases with increasing the excitation pulse carrier frequency) and negative (opposite trend). In the area of low exergonicity of the photoinduced charge transfer, the effect is mostly positive, while in the field of strong exergonicity, it is mostly negative. The amplitude of the vibrational spectral effect predicted by the model is rather large and can be observed in experiments even if the vibrational relaxation/redistribution time constant is as short as 100 fs.

I. INTRODUCTION Photoinduced charge transfer reactions in donor−acceptor pairs involve at least three stages and may be described as follows:

postulates independence of the quantum yield of luminescence from the wavelength of exciting radiation. However, the charge transfer can be so fast that can effectively compete with the nuclear relaxation and violate the Kasha−Vavilov rule.3−9 Ultrafast photochemical processes in excited molecular systems occur on time scales comparable with the relaxation times of the nuclear subsystem of the reactants and solvent.10−13 This means that the nonequilibrium of the nuclear subsystem created by pump pulse is kept during the photochemical conversion and, therefore, may manifest in the dynamics of populations of the reactants and products, including intermediates. Since the degree of nonequilibrium of the nuclear subsystem is defined by spectral characteristics of the pump pulse (carrier frequency, duration, etc.), it becomes possible to control the rate of photochemical reactions and product yields by varying the characteristics of the pump pulse. The spectral effect (a dependence of the charge transfer rate constant on the excitation pulse carrier frequency) was observed in ultrafast charge recombination of excited donor− acceptor complexes in polar solvents.4,5,8 In these systems, the charge recombination was supposed to proceed from a nonequilibrium state of the surrounding medium created by the pumping pulse. Supposing the relaxation of intramolecular high frequency vibrational modes to occur at considerably

Here ωe is the carrier frequency of the excitation pulse, kCS (two arrows pointing in opposite directions indicate the reversibility of the transitions between D*A and D+A− states) and kCR are the charge separation (CS) and recombination rate constants, correspondingly, kD is the total rate constant of radiative and nonradiative transitions in the excited donor D*. A short pumping pulse transfers the donor−acceptor pair from the ground to an excited Franck−Condon state in which the surrounding medium and intramolecular vibrational degrees of freedom can be far from their thermal equilibrium. At the next stage, the nuclear relaxation (medium relaxation and intramolecular vibrational relaxation/redistribution) proceeds in competition with the charge transfer, hot luminescence and internal conversion. Typically, the nuclear relaxation is much faster than the competing chemical processes that manifests itself as independence of the charge transfer kinetics from the excitation pulse carrier frequency.1,2 This conclusion is a direct consequence of the well-known Kasha−Vavilov rule that © XXXX American Chemical Society

Received: January 1, 2015 Revised: January 27, 2015

A

DOI: 10.1021/acs.jpcc.5b00005 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

transfer dynamics on details of solvation dynamics.8,9,17,19−22 The electron−vibration interaction plays a key role in the vibrational spectral effect formation studied here, since this interaction is responsible for the activity of high frequency vibrational modes on both the excitation step, and the charge transfer stage. Figure 1 shows a few of possible states produced

shorter time scales than the medium relaxation and charge recombination, the spectral effect was rather well described in the framework of the stochastic multichannel point-transition model.8 The nonequilibrium of the intramolecular high frequency modes produced by the excitation pulse may also manifest in charge transfer kinetics if the charge transfer rate constant is comparable to the intramolecular vibrational redistribution rate. Manifestations of the excited states of high-frequency vibrations in the charge transfer dynamics were reported in a number of experimental studies.3,6,7,14−16 Moreover, analysis of experimental kinetic data has showed that the rate of photoinduced electron transfer from the vibrational ground state and the first excited state (vibrational mode with the frequency of 0.17 eV) in porphyrin-bridge−quinone dyad differ considerably.7 Theoretical calculations are in good agreement with these experimental estimates.17 This result indicates that vibrational relaxation time scale can be comparable to or exceed the photoinduced charge transfer time constant. The molecules of interest to the study of ultrafast charge transfer have dozens or even hundreds of vibrational degrees of freedom. However, only a small part of them is active (can be excited) at the stage of photoexcitation. Further, in the charge transfer stage, there is also a small number of accepting vibrational modes. To observe the vibrational spectral effect, at least one vibrational mode has to be simultaneously active in both stages. At the present time experimental information on the activity of the vibrational modes at the charge transfer stage is practically absent. There are only estimations of the total reorganization energy of high-frequency modes that lies in the range 0.2−0.4 eV.18 Making use of simple combinatorial evaluations one can find that the probability of a given vibrational mode to be the active in both stages is very small. Obviously, there are a few molecular systems that can demonstrate the vibrational spectral effect. According to indirect experimental data in the porphyrin-bridge-quinone dyad a vibrational mode of the porphyrin ring with a frequency of 0.17 eV is active in both stages. This means that the vibrational spectral effect can be expected in molecular systems involving porphyrin derivatives as acceptor or donor. Since the porphyrin-based compounds are widely used at present in connection with the problems of photosynthesis, organic photovoltaics et al., the study of vibrational spectral effect in such systems may be of great interest. The goals of this paper are (i) to develop a model of photoinduced intramolecular charge transfer involving both excitation and charge transfer stages, (ii) to estimate the magnitude of the vibrational spectral effect caused by excitation of the intramolecular high frequency vibrational modes and its dependencies on the charge transfer parameters, and (iii) to reveal the parameter regions where the vibrational spectral effect is large enough to be observable in experiments.

Figure 1. Scheme of the electronic states involved in photoinduced electron transfer with a few vibrational sublevels. The vertical arrows indicate the photoexcitation at different wavelengths, the vertical wavy lines visualize the intramolecular vibrational relaxation/redistribution, the solvent relaxation and electronic transition are shown with short arrows. The filled bells visualize a part of the ground state population transferred to certain sublevels of the excited electronic state. The bells pictured with dashed lines point out locations of the initially created bells some time later which they have occupied due to electronic transitions and the solvent relaxation.

by excitation pulse and possible routes of their evolution: charge transfer from vibrational unrelaxed states proceeds in competition with its relaxation (vertical transitions) and motion of the wave packets to the equilibrium position (medium relaxation). The vibrational spectral effect can be observed if vibrational relaxation is sufficiently slow and charge transfer can compete with it. One can expect that a typical time scale of these processes is several hundreds of femtoseconds. To observe such charge transfer kinetics the pumping and probing pulses should have duration of the order of tens of femtoseconds. In this case the pump and charge transfer stages practically do not overlapped. As a result, one may first calculate the population distribution in a locally excited state produced by a pumping pulse and then use it as the initial condition for photoinduced charge transfer. In what follows, this approach is applied. Dynamical properties of real polar solvents usually are characterized by a nonexponential relaxation function.23−26 In such solvents charge transfer are generally described in terms of multidimensional free energy surfaces corresponding to different electronic and vibronic states participating in the reaction. The relaxation function of a solvent, X(t), can be represented as a sum of several (N) exponential summands:24,26

II. STOCHASTIC MODEL OF PHOTOINDUCED CHARGE TRANSFER INVOLVING THE EXCITATION STAGE According to eq 1, the model should include at least three electronic states: the ground state, a locally excited state, and a charge separated state. Interaction of the transferred charge with a polar solvent is of primary importance for charge transfer kinetics and is also an integral part of the model. This interaction essentially determines the energetic parameters of the electronic transitions. Besides, there is dynamic solvent effect that reveals itself through strong dependence of charge

N

X (t ) =

N

∑ xie−t /τ ∑ xi = 1 i

i=1

i=1

(2)

Each summand is associated with a separate solvent coordinate Qi with relaxation time constant τi. The solvent coordinates B

DOI: 10.1021/acs.jpcc.5b00005 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C form N-dimensional configuration space of the considered problem. The quantities Qi being collective degrees of freedom of a solvent are usually associated with different kinds of relaxation processes of the medium. The profiles of the Gibbs free energy along the reaction coordinate Q are pictured in Figure 1. The diabatic free energy surfaces for the ground state, UGS, locally excited state with the vibrationally excited sublevels U(n) LES, and the charge separated electronic state U(m) in terms of the reaction coordinates Qi(i = CS 1, .., N) can be written as follows21

(m) UCS =

2Eriex )2

∑ (Q i −

+ ΔGGS

1 2

(3)

1 2

(4)

i=1

2EriCS )2 + mℏΩ + ΔGCS

i=1

∂t − (m) ∂ρCS

∂t −

(n) = L̂ LESρLES −

(n) (m) )+ − ρCS ∑ knm(ρLES m

(5)

n (m) ρCS τv(m)

N

∑ i=1

∑ ∫ ρCS(m)(Q, t ) ∏ dQ i i=1

m

(12)

(13)

To specify the initial conditions, we assume that the pump pulse has a Gaussian form ⎧ t2 ⎫ E(t ) = E0 exp⎨iωet − 2 ⎬ τe ⎭ ⎩

(14)

and its duration is short enough so that the medium is considered to be frozen during excitation. All high frequency vibrational modes are supposed initially to be in the ground state. This allows us to obtain the following general expression for the initial probability distribution function on the excited term4,28 2 ⎧ (n) ⎪ ∑ Q̃ i 2Eriex ]τe 2 [ ℏ δω − e (n) (Q, t = 0) = APn exp⎨− ρLES 2ℏ2 ⎪ ⎩

τv(n + 1)

(m + 1) ρCS

−

τv(m + 1)

Pn =

(7)

∂ ∂ 2 ⎤⎥ 1 ⎡⎢ + kBT 1 + Qi τi ⎢⎣ ∂Q i ∂Q i 2 ⎥⎦

i=1 N

PCS(t ) =

where Q is a vector with components Q1, Q2, ..., QN, L̂ LES, L̂ CS are the Smoluchowski operators describing diffusion on the (m) U(n) LES and UCS free energy surfaces, correspondingly, L̂ LES =

(n) (Q, t ) ∏ dQ i ∑ ∫ ρLES n

(6) (n) (m) )+ − ρCS ∑ knm(ρLES

(10)

N

(n) ρLES

(m) = L̂CSρCS +

2πV 2Fnm (n) (m) δ(ULES − UCS ) ℏ

PLES(t ) =

(n + 1) ρLES

τv(n)

(9)

where S = Erv/ℏΩ and Erv are the Huang−Rhys factor and the reorganization energy of the intramolecular high frequency mode, Fnm is the Franck−Condon factor for the transition between the vibrational sublevels n and m. The populations of the excited electronic state and the charge separated state are calculated with the equation

where ΔGGS is the free energy change for electronic transition from locally excited to the ground state, ΔGCS is the free energy change of the photoinduced charge separation, n and m are the quantum numbers of intramolecular high frequency vibrational mode with the frequency Ω that must be Ω ≫ kBT with kB and T being the Boltzmann constant and the temperature, Ees ri and ECS ri are the reorganization energies of the i-th medium mode of the solvent at the stages of pump and charge separation, correspondingly. It is related to its weight in the relaxation ex CS CS ex ex CS function eq 2 by xi = Eex ri /Erm = Eri /Erm , Erm = ∑i Eri and Erm = ∑i ECS are the total reorganization energies of the solvent at ri corresponding stage. The reorganization energy for excitation stage is typically considerably smaller than that of the charge transfer stage and reveals itself mainly in the position and the form of the wave packet on the free energy surface of the locally excited state. Making use of the stochastic point-transition approach,8,20,21 the temporal evolution of the system is described by a set of equations for the probability distribution functions for the locally excited electronic states ρ(n) LES(Q, t) and the charge separated state ρ(m) CS (Q, t) (n) ∂ρLES

∂ ∂ 2 ⎤⎥ + kBT ∂Q i ∂Q i 2 ⎥⎦

(11)

N

∑ (Q i −

2EriCS )

⎡ min(n , m) ⎤2 ( −1)m − r ( S )n + m − 2r ⎥ ⎢ Fnm = exp{−S}n! m ! ∑ ⎢⎣ r = 0 r! (n − r ) ! (m − r )! ⎥⎦

N

∑ Q i2 + nℏΩ

1 ⎡⎢ 1 + (Q i − τi ⎢⎣

The model, eqs 6 and 7, implies a single-quantum mechanism of the irreversible vibrational relaxation of the type n → n − 1 27 (n) (n) with the rate constant (1/(τ(n) v )), where τv = ((τv )/n). Transitions between vibrational sublevels of the locally excited electronic state U(n) LES and of the charge separated state U(m) CS are described by the Zusman parameters knm =

i=1

∑ i=1

N

1 UGS = 2 (n) ULES =

N

̂ = LCS

∑

⎫ 2Eriex )2 ⎪ ⎬ 2kBT ⎪ ⎭

(Q i −

(15)

⎧ [ℏδω(n)]2 ⎫ S n e −S e ⎬ exp⎨− n! 2σ 2 ⎭ ⎩ ⎪

⎪

⎪

⎪

(16)

2 ex ex where ℏδω(n) e = −Erm + ΔGGS − nℏΩ + ℏωe, σ = (2ErmkBT) + ℏ2τe−2, and Pn is a factor proportional to the fraction of the excited donor−acceptor pairs in the vibrational state with vibrational quantum number n. The factor A depends on the power and duration of the pump pulse and determined by eq 17

(8) C

DOI: 10.1021/acs.jpcc.5b00005 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C We =

(n) (Q, t = 0) ∏ dQ i ∫ ∑ ρLES i

n

compound a vibrational mode of the porphyrin ring with a frequency of 0.17 eV is rather long-lived and its relaxation time is comparable with photoinduced charge separation time scale. This is the reason why the parameters used further are altered around those obtained in the experiment and reported in ref 7. Dynamical characteristics of acetonitrile are exploited for the description of the solvent dynamics: weights of two Debye’s modes x1 = 0.686, x2 = 0.314 and relaxation times τ1 = 0.089 ps, τ2 = 0.63 ps.32 In Figure 2, the dependence of the vibrational spectral effect on the solvent reorganization energy, ECS rm , for three values of

(17)

where We is the probability of the electronic excitation by the pump pulse. The initial conditions for the charge separated state distribution functions, ρ(m) CS (Q, t) are (m) ρCS (Q, t = 0) = 0

(18)

The set of eqs 6 and 7 with the initial conditions eq 15− 18 is solved numerically by using the Brownian simulation method.4,29,30 The model accounts for excited vibrational states with short lifetime. Their relaxation reveals itself not only through variation of population of the vibrational states but also through its influence on the rates of the transitions between vibrational sublevels of the locally excited state and the charge separated state. The mechanism of this influence is discussed in detail in ref 31. In the conclusion of this section we briefly summarize the physical processes incorporated into the model. The excitation of the system by a pumping pulse is visualized as an appearance of a single wave packet on a sublevel of the locally excited state 1/2 if inequality ℏΩ ≫ (2kBTEex is fulfilled. In the opposite rm) limit several wave packets simultaneously appear on different sublevels of the excited state (see Figure 1). Further three competing processes proceed. The first is the vibrational relaxation n → n − 1 with the time constant τ(n) v , the second is the wave packet motion to the bottom of the free energy surface U(n) LES reflecting the medium relaxation, and the third is the charge separation occurring at the term crossing points that populates the vibrational sublevels of the charge separated state (dashed lines in Figure 1). Next, the systems created in the charge separated state at the term crossing points move to the (m) UCS term minimum undergoing, in parallel, quantum (m) transitions between neighbor vibrational states UCS → U(m−1) . All these processes approach the system to its thermal CS equilibrium that is the final point of its evolution. Here only ultrafast charge separation occurring in femto- and picosecond area is considered. Since the fluorescence and internal conversion from the locally excited state proceed much slower than the charge separation they are omitted from the consideration in this paper.

Figure 2. Dependence of the vibrational spectral effect, χ = [k(ωe = 2.37) − k(ωe = 2.20)]/k(ωe = 2.20), on ECS rm : (1) ΔGCS = −0.4 eV (black), (2) ΔGCS = −0.8 eV (red), (3) ΔGCS = −1.2 eV (blue); ΔGGS = −2.1 eV; Erv = 0.3 eV, Eex rm = 0.1 eV; VCS = 0.05 eV, τv = 1.0 ps.

the reaction free energy, ΔGCS is pictured. The magnitudes of all parameters are listed in the figure caption. Two excitation frequencies, ℏωe = 2.20 and 2.37 eV, are adopted because they correspond to the maximum absorption bands of the transitions to the ground and first excited vibrational states. These bands weakly overlap due to small solvent reorganization energy at the excitation stage, Eex rm. To gain an insight into physical mechanisms underlying the dependencies pictured in Figure 2 let us first consider the low exergonic reaction (ΔGCS = −0.4 eV). Here the inequality −ΔGCS < ECS rm is fulfilled. Obviously, for such parameters the charge separated state is predominantly formed in the ground vibrational state due to smaller activation energy (see Figure 1 that corresponds to this parameters). In this case the rate of charge separation from the first excited vibrational state S11 is larger than that from the ground state S10 due to two reasons: (i) the Franck−Condon factor is larger for vibrational excited state, F10/F00 = S > 1; (ii) the activation barrier for charge separation from S11 state is smaller. This results in a positive vibrational spectral effect. The spectral effect raises with increasing ECS rm because the difference between the heights of the activation barrier increases. In other words k(ωe2) decreases with increasing ECS rm slower than k(ωe1). This trend is turned over when the charge separation rate becomes smaller than the vibrational relaxation rate, 1/τv. Indeed, for the ECS rm = 0.65 eV the rate constant k(ωe1) = 1.03 ps−1. For very large ECS rm the charge separation occurs from the vibrational ground state independently form excitation frequency and the effect approaches to zero.

III. VIBRATIONAL SPECTRAL EFFECT IN PHOTOINDUCED CHARGE TRANSFER KINETICS In this section the influence of carrier frequency of the excitation pulse on the rate of photoinduced charge transfer is investigated. For quantitative description of the phenomenon a vibrational spectral effect is exploited χ=

k(ωe2) − k(ωe1) k(ωe1)

(19)

Here the rate k(ωe) is determined by eq 20 1 = k(ωe)

∫0

∞

dt PLES(t )

(20)

and both frequencies ωe1 and ωe2 are supposed to fall in an absorption band corresponding to a single electronic transition. To adopt realistic parameters we refer to Zn−porphyrinbridge−quinone dyad in which the intramolecular photoinduced charge separation from both the vibrationally unrelaxed and the relaxed states has been observed.7 In this D

DOI: 10.1021/acs.jpcc.5b00005 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C In the Marcus inverted region, ΔGCS = −1.2 eV, the charge separation mainly proceeds through the sinks placed at the bottom of the free energy surface U(n) LES. In this case the ratio of the charge separation rate constants form the excited, k1, and the ground vibrational states, k0, is determined by the ratio of corresponding Franck−Condon factors

2). Increasing the effect with Erv for low exergonic reaction, ΔGCS = −0.4 eV is a consequence of increasing the ratio

F10 E = rv F00 ℏΩ

This is the only reason since the activation barrier is unchangeable. The relatively small slope of the black curve in Figure 3 reflects a considerably weaker dependence of the ratio k1/k0 on Erv than that expected in nonadiabatic regime where k1/k0 = F10/F00. Such a weakening of the dependence is a direct consequence of the manifestation of the dynamic solvent effect.9,20 In the case of large exergonicity, ΔGCS = −1.2 eV, decreasing the effect with Erv can be deduced from eq 21. This equation predicts the negative effect in the range n* − (n*)1/2 < S < n* + (n*)1/2 that can be rewritten in terms of the vibrational reorganization energy 0.25 < Erv < 0.88 eV for the accepted parameters. The simulations show the negative effect in the range Erv > 0.21 eV that is in agreement with the estimation. A few dependencies of the vibrational spectral effect on the vibrational relaxation time scale, τv, are pictured in Figure 4. It

2

F k1 (n* − S) ∼ 1n * = k0 F0n *−1 n*

(21)

where n* =

CS −ΔGCS − Erm + ℏΩ ℏΩ

(23)

(22)

is the number of the sink which is the nearest to the minimum of U(1) LES. Equation 21 predicts decreasing the spectral effect with CS increasing ECS rm . The area of Erm where the negative effect is expected can be find out from the condition k1/k0 < 1. From eq 21 one obtains 0.84 < n* < 3.68 or 0.74 < ECS rm < 1.23 eV. In the simulations the negative effect is placed at ECS rm > 0.65 eV. Accounting for roughness of the estimation, the predictions are in accord with the simulations. It should be noticed that eq 21 is applicable only in the nonadiabatic regime. Its agreement with the simulations allows us to conclude that despite large value of electronic coupling, V, the reaction in the inverted region proceeds in the nonadiabatic regime. This is due to reduction of the effective electronic coupling by Franck− Condon factors that for the most effective sinks are rather small (see eq 10).9 The case with ΔGCS = −0.8 eV (red line in Figure 2) is intermediate between two considered above. In the region of small reorganization energy ECS rm , the charge separation proceeds in the Marcus inverted region and decreasing the spectral effect with increase of ECS rm is observed. In the region of large values of ECS rm , the parameters approach to the Marcus normal region where the opposite trend is realized. In Figure 3, the dependence of the vibrational spectral effect on the vibrational reorganization energy, Erv, is presented. The curves in this figure looks similar to those presented in Figure

Figure 4. Dependence of the vibrational spectral effect on the vibrational relaxation time scale, τv. The parameters are (1) ΔGCS = −0.4 eV (black), (2) ΔGCS = −0.8 eV (red), (3) ΔGCS = −1.2 eV ex (blue); Erv = 0.3 eV, ECS rm = 0.8 eV, Erm = 0.1 eV, VCS = 0.05 eV.

demonstrates expected increase of the amplitude of the vibrational spectral effect with increasing τv. Indeed, supposing the populations of the first excited, S11, and the ground, S10, vibrational states to decay (due to charge transfer) exponentially with the rate constants k1 and k0, correspondingly, one can derive the dependence of the spectral effect on the vibrational relaxation time constant: χ=

(k1 − k 0)τv 1 + k 0τv

(24)

This equation predicts a monotonous raise of the absolute value of the spectral effect with increasing τv. The dependence is saturated when the inequality k0τv ≫ 1 is fulfilled. For ΔGCS = −0.8 and ΔGCS = −1.2 eV the rates k0 ≃ k(ωe = 2.20) are equal to 7.9 ps−1 and 15.4 ps−1, correspondingly and the saturation is achieved at τv ≈ 1 ps. For ΔGCS = −0.4 eV the rate k0 = 0.34 ps−1 and the saturation is expected at τv ≈ 30 ps. So

Figure 3. Dependence of the vibrational spectral effect, χ = [k(ωe = 2.37) − k(ωe = 2.20)]/k(ωe = 2.20), on the vibrational reorganization energy, Erv. The parameters are (1) ΔGCS = −0.4 eV (black), (2) ΔGCS = −0.8 eV (red), (3) ΔGCS = −1.2 eV (blue); ΔGGS = −2.1 eV; ex ECS rm = 0.8 eV, Erm = 0.1 eV; VCS = 0.05 eV, τv = 1.0 ps. E

DOI: 10.1021/acs.jpcc.5b00005 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

it is mostly negative. In the barrierless region a weak effect is expected. The amplitude of the vibrational spectral effect predicted by the model can be rather large to be easily observed in experiments even if the vibrational relaxation time, τv, is as short as 100 fs. There are at least two mechanisms suppressing the effect. The first is very fast redistribution/relaxation of excited vibrational modes. One of possible ways to enlarge the relaxation time is decreasing the temperature. The results reported in ref 7 give evidence that at low temperature the vibrational relaxation can be noticeable slower than the intramolecular charge transfer. Thus, the decrease in temperature may increase the effect, making it much more pronounced. The second reason is connected with a large number of vibrational degrees of freedom in the molecules of interest. The Franck−Condon active modes at the stage of excitation can be inactive at the stage of charge transfer. In this case full suppression of the vibrational spectral effect is expected (a very small effect can be due to heating of a molecule). This strongly limits the circle of polyatomic molecules in which the spectral effect can be observed. The way out of this situation could be associated with usage of a synchronous excitation of the electronic and vibrational transitions. This type of experiment could figure out which vibrational modes are active at the stage of the charge transfer. Moreover, the synchronous IR pumping can provide a tool of control of photoinduced charge transfer rate.

that the spectral effect in the Marcus normal region can be very large for slowly relaxing vibrational modes. Another important result is a prediction of an observable effect at τv as short as 100 fs that corresponds to typical values of the vibrational relaxation time constant. It should be noted that eq 24 does not account for an important process which takes into account the stochastic model. This is charge separation from vibrationally excited states in the Marcus normal region that is accompanied by hot recombination into lower lying vibrational sublevels of the locally excited state. The probability of such hot recombination for ultrafast reaction can be as large as few tens of percents.33 Obviously, this indirect population of lower lying vibrational states considerably increases the effectiveness of the vibrational relaxation and decreases the vibrational spectral effect. Dependence of the vibrational spectral effect on the excitation pulse carrier frequency, ωe, is pictured in Figure 5.

■

AUTHOR INFORMATION

Corresponding Author

*(A.I.I.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (Grants No. 14-03-00261 and No 14-03-97044).

Figure 5. Dependence of the vibrational spectral effect, χω = [k(ωe) − k(ωe = 2.20)]/k(ωe = 2.20), on the excitation pulse carrier frequency, ωe. The parameters are (1) ΔGCS = −0.4 eV (black), (2) ΔGCS = −0.8 eV (red), (3) ΔGCS = −1.2 eV (blue); ΔGGS = −2.1 eV, ECS rm = 0.8 eV, Erv = 0.3 eV, Eex rm = 0.8 eV, VCS = 0.05 eV, τv = 1.0 ps.

■

REFERENCES

(1) Elsaesser, T.; Kaiser, W. Vibrational and Vibronic Relaxation of Large Polyatomic Molecules in Liquids. Annu. Rev. Phys. Chem. 1991, 42, 83−107. (2) Kovalenko, S. A.; Schanz, R.; Hennig, H.; Ernsting, N. P. Cooling Dynamics of an Optically Excited Molecular Probe in Solution from Femtosecond Broadband Transient Absorption Spectroscopy. J. Chem. Phys. 2001, 115, 3256−3273. (3) Zimmermann, C.; Willig, F.; Ramakrishna, S.; Burfeindt, B.; Pettinger, B.; Eichberger, R.; Storck, W. Experimental Fingerprints of Vibrational Wave-Packet Motion during Ultrafast Heterogeneous Electron Transfer. J. Phys. Chem. B 2001, 105, 9245−9253. (4) Fedunov, R. G.; Feskov, S. V.; Ivanov, A. I.; Nicolet, O.; Pagès, S.; Vauthey, E. Effect of the Excitation Pulse Carrier Frequency on the Ultrafast Charge Recombination Dynamics of Donor-Acceptor Complexes: Stochastic Simulations and Experiments. J. Chem. Phys. 2004, 121, 3643−3656. (5) Nicolet, O.; Banerji, N.; Pagès, S.; Vauthey, E. Effect of the Excitation Wavelength on the Ultrafast Charge Recombination Dynamics of Donor−Acceptor Complexes in Polar Solvents. J. Phys. Chem. A 2005, 109, 8236−8245. (6) Wan, C.; Xia, T.; Becker, H. C.; Zewail, A. H. Ultrafast Unequilibrated Charge Transfer: A New Channel in the Quenching of Fluorescent Biological Probes. Chem. Phys. Lett. 2005, 412, 158−163. (7) Kang, Y. K.; Duncan, T. V.; Therien, M. J. TemperatureDependent Mechanistic Transition for Photoinduced Electron Trans-

Here the effect is determined as follows: χω = [k(ωe) − k(ωe = 2.20)]/k(ωe = 2.20). The simulations demonstrate that the vibrational spectral effect monotonously enhances with increasing the excitation frequency. For low exergonic reactions (ΔGCS = −0.4 eV) this trend is a direct consequence of reducing the height of the activation barrier with increasing the number of the excited vibrational state. For the reactions in the inverted region (ΔGCS = −1.2 eV) the trend reflects a reduction of the Franck−Condon factors with increasing the number of the excited vibrational state.

IV. CONCLUDING REMARKS A model of photoinduced ultrafast charge transfer involving an excited state formation by a short laser pulse is developed. The model demonstrates a rather strong dependence of the photoinduced charge transfer rate constant on excitation pulse carrier frequency. To quantify this dependence the vibrational spectral effect is introduced. The simulations show that the spectral effect may be both positive and negative. In the field of low exergonicity (−ΔGCS < ECS rm ) the effect is mostly positive while in the field of strong exergonicity (−ΔGCS > ECS rm ) F

DOI: 10.1021/acs.jpcc.5b00005 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C fer Modulated by Excited-State Vibrational Relaxation Dynamics. J. Phys. Chem. B 2007, 111, 6829−6838. (8) Feskov, S. V.; Ionkin, V. N.; Ivanov, A. I.; Hagemann, H.; Vauthey, E. Solvent and Spectral Effects in the Ultrafast Charge Recombination Dynamics of Excited Donor-Acceptor Complexes. J. Phys. Chem. A 2008, 112, 594−601. (9) Yudanov, V. V.; Mikhailova, V. A.; Ivanov, A. I. Reorganization of Intramolecular High Frequency Vibrational Modes and Dynamic Solvent Effect in Electron Transfer Reactions. J. Phys. Chem. A 2012, 116, 4010−4019. (10) Heitele, H.; Pöllinger, F.; Häberle, T.; Michel-Beyerle, M. E.; Staab, H. A. Energy Gap and Temperature Dependence of Photoinduced Electron Transfer in Porphyrin-Quinone Cyclophanes. J. Phys. Chem. 1994, 98, 7402−7410. (11) Akesson, E.; Walker, G. C.; Barbara, P. F. Dynamic Solvent Effects on Electron Transfer Rates in the Inverted Regime: Ultrafast Studies on the Betaines. J. Chem. Phys. 1991, 95, 4188−4194. (12) Kandori, H.; Kemnitz, K.; Yoshihara, K. Subpicosecond Transient Absorption Study of Intermolecular Electron Transfer between Solute and Electron-Donating Solvents. J. Phys. Chem. 1992, 96, 8042−8048. (13) Seel, M.; Engleitner, S.; Zinth, W. Wavepacket Motion and Ultrafast Electron Transfer in the System Oxazine 1 in N,Ndimethylaniline. Chem. Phys. Lett. 1997, 275, 363−369. (14) Lenzmann, F.; Krueger, J.; Burnside, S.; Brooks, K.; Grätzel, M.; Gal, D.; Ruëhle, S.; Cahen, D. Surface Photovoltage Spectroscopy of Dye-Sensitized Solar Cells with TiO2, Nb2O5, and SrTiO3 Nanocrystalline Photoanodes: Indication for Electron Injection from Higher Excited Dye States. J. Phys. Chem. B 2001, 105, 6347−6352. (15) Shaw, L. E.; Langford, C. H. The photochemistry of Mo(CNPh)6: Dissociative Photosubstitution and Evidence for Hot Electron Transfer. Coord. Chem. Rev. 2002, 230, 165−169. (16) Kallioinen, J.; Benkö, G.; Sundström, V.; Korppi-Tommola, J. E. I.; Yartsev, A. P. Electron Transfer from the Singlet and Triplet Excited States of Ru(dcbpy)2(NCS)2 into Nanocrystalline TiO2 Thin Films. J. Phys. Chem. B 2002, 106, 4396−4404. (17) Kichigina, A. O.; Ionkin, V. N.; Ivanov, A. I. U-Shaped Temperature Dependence of Rate Constant of Intramolecular Photoinduced Charge Separation in Zinc Porphyrin Bridge Quinone Compounds. J. Phys. Chem. B 2013, 117, 7426−7435. (18) Gould, I. R.; Young, R. H.; Moody, R. E.; Farid, S. Contact and Solvent-Separated Geminate Radical Ion Pairs in Electron-Transfer Photochemistry. J. Phys. Chem. 1991, 95, 2068−2080. (19) Yakobson, B. I.; Burshtein, A. I. Relaxation Hindrance in Nonadiabatic Cage Reactions. Chem. Phys. 1980, 49, 385−395. (20) Zusman, L. D. Outer-Sphere Electron Transfer in Polar Solvents. Chem. Phys. 1980, 49, 295−304. (21) Feskov, S. V.; Ionkin, V. N.; Ivanov, A. I. Effect of HighFrequency Modes and Hot Transitions on Free Energy Gap Dependence of Charge Recombination. J. Phys. Chem. A 2006, 110, 11919−11925. (22) Yudanov, V. V.; Mikhailova, V. A.; Ivanov, A. I. Nonequilibrium Phenomena in Charge Recombination of Excited Donor-Acceptor Complexes and Free Energy Gap Law. J. Phys. Chem. A 2010, 114, 12998−13004. (23) Rosenthal, S. J.; Xie, X.; Du, M.; Fleming, G. R. Femtosecond Solvation Dynamics in Acetonitrile: Observation of the Inertial Contribution to the Solvent Response. J. Chem. Phys. 1991, 95, 4715−4718. (24) Jimenez, R.; Fleming, G. R.; Kumar, P. V.; Maroncelli, M. Femtosecond Solvation Dynamics of Water. Nature 1994, 369, 471− 473. (25) Gumy, J. C.; Nicolet, O.; Vauthey, E. Investigation of the Solvation Dynamics of an Organic Dye in Polar Solvents Using the Femtosecond Transient Grating Technique. J. Phys. Chem. A 1999, 103, 10737−10743. (26) Maroncelli, M.; Kumar, V. P.; Papazyan, A. A Simple Interpretation of Polar Solvation Dynamics. J. Phys. Chem. 1993, 97, 13−17.

(27) Ivanov, A. I.; Ionkin, V. N.; Feskov, S. V. Acceleration of the Recombination of Photoexcited Donor-Acceptor Complexes with a High-Frequency Vibrational Mode. Russ. J. Phys. Chem. A 2008, 82, 303−309. (28) Ivanov, A. I.; Belikeev, F. N.; Fedunov, R. G.; Vauthey, E. The Effect of Excitation Pulse Carrier Frequency on Ultrafast Charge Recombination Dynamics of Excited Donor-Acceptor Complexes. Chem. Phys. Lett. 2003, 372, 73−81. (29) Ionkin, V. N.; Ivanov, A. I. Independence of the Rate of the Hot Charge Recombination in Excited Donor-Acceptor Complexes from the Spectral Density of High-Frequency Vibrations. Chem. Phys. 2009, 360, 137−140. (30) Gladkikh, V.; Burshtein, A. I.; Feskov, S. V.; Ivanov, A. I.; Vauthey, E. Hot Rrecombination of Photogenerated Ion Pairs. J. Chem. Phys. 2005, 123, 244510−1−11. (31) Mikhailova, V. A.; Ivanov, A. I. Effect of Relaxation of Intramolecular High-Frequency Vibrational Mode on Nonthermal Electron Transfer Probability. Stochastic Point-Transition Approach. J. Phys. Chem. C 2007, 111, 4445−4451. (32) Horng, M. L.; Gardecki, J. A.; Papazyan, A.; Maroncelli, M. Subpicosecond Measurements of Polar Solvation Dynamics − Coumarin 153 Revisited. J. Phys. Chem. 1995, 99, 17311−17337. (33) Ivanov, A. I.; Potovoi, V. V. Theory of Non-Thermal Electron Transfer. Chem. Phys. 1999, 247, 245−259.

G

DOI: 10.1021/acs.jpcc.5b00005 J. Phys. Chem. C XXXX, XXX, XXX−XXX