Nonequilibrium Phenomena in Charge Recombination of Excited

12998

J. Phys. Chem. A 2010, 114, 12998–13004

Nonequilibrium Phenomena in Charge Recombination of Excited Donor-Acceptor Complexes and Free Energy Gap Law Vladislav V. Yudanov, Valentina A. Mikhailova, and Anatoly I. Ivanov* Department of Physics, Volgograd State UniVersity, UniVersity AVenue 100, Volgograd 400062, Russia ReceiVed: August 3, 2010; ReVised Manuscript ReceiVed: October 29, 2010

The charge recombination dynamics of excited donor-acceptor complexes in polar solvents has been investigated within the framework of the stochastic approach. The model involves the excited state formation by the pump pulse and accounts for the reorganization of a number of intramolecular high-frequency vibrational modes, for their relaxation as well as for the solvent reorganization following nonexponential relaxation. The hot transitions accelerate the charge recombination in the low exergonic region and suppress it in the region of moderate exothermicity. This straightens the dependence of the logarithm of the charge recombination rate constant on the free energy gap to the form that can be fitted to the experimental data. The free energy dependence of the charge recombination rate constant can be well fitted to the multichannel stochastic model if the donor-acceptor complexes are separated into a few groups with different values of the electronic coupling. The model provides correct description of the nonexponential charge recombination dynamics in excited donor-acceptor complexes, in particular, nearly exponential recombination in perylene-tetracyanoethylene complex in acetonitrile. It appears that majority of the initially excited donor-acceptor complexes recombines in a nonthermal (hot) stage when the nonequilibrium wave packet passes through a number of term crossings corresponding to transitions toward vibrational excited states of the electronic ground state in the area of the low and moderate exothermicity. I. Introduction Photoexcitation of donor-acceptor complexes (DAC) by a short laser pulse in charge transfer band triggers a series of processes in accord with scheme: φ

D+A- 98 D+ + Ahν vV kCR DA where hν indicates the photoexcitation of the ground-state complex, DA, leading to the population of an excited state that is a contact ion-radical pair, D+ A-, kCR is the rate constant of intermolecular charge recombination (CR), and φ is the free ion quantum yield. Geminate CR in excited DACs is surely the most investigated chemical reaction. Nevertheless, the mechanism of CR is still not completely clarified. The experimental investigations of CR kinetics in excited DACs demonstrated unexpected dependence of the CR rate constant on the reaction free energy in the weakly exergonic region.1,2 Namely, the logarithm of the CR rate constant decreases monotonically, nearly linearly, with increasing the reaction exothermicity, -∆GCR,1,2 whereas the standard equilibrium Marcus nonadiabatic theory predicts a bell-shaped dependence.3 A few mechanisms have been proposed to account for such a behavior of the rate constant. According to ref 2, CR kinetics cannot be explained in the framework of the Marcus nonadiabatic theory and, hence, alternative mechanisms should be invoked. It was supposed that CR in the excited DACs is * To whom correspondence should be addressed. E-mail: Anatoly.Ivanov@ volsu.ru.

associated with reorganization of intramolecular and intracomplex vibrational high-frequency modes and that the solvent plays a minor role.2 This means that theory of nonradiative transitions in polyatomic molecules predicting nearly linear dependence of the logarithm of the rate constant on the driving force is applicable for description of CR kinetics. Another explanation of such a dependence of the rate is based on the fact that the laser pulse populates a nonequilibrium initial vibrational state of the DAC, proposed in ref 4. Although the calculations performed within the framework of the stochastic point-transition approach5,6 could well reproduce the experimentally observed free energy dependence of the rate, the explanation encountered two problems.7 A good fit requires too large electronic coupling values and besides the model predicts a strong time dependence of the rate constant, in disagreement with most experimental data.2,8-10 A later model considering the CR of excited DACs as a transition between excited and ground adiabatic states induced by the nonadiabatic interaction has been investigated in ref 7. Assuming that the reaction proceeds in stationary regime, the authors could very well reproduce the experimentally observed free energy dependence of the rate constant. However, the typical solvent relaxation time scale are often of the same order or even shorter than that of CR. Therefore, a model predicting a consistent description of ultrafast electron transfer dynamics has to include explicit consideration of the nuclear relaxation. A generalization of this model to the nonequilibrium regime also has revealed a rather strong time dependence of the CR rate constant.11 One more weakness of this approach is the prediction of a decrease of the excited state ionicity in the course of the medium relaxation. According to this theory, the adiabatic states, Ψ(, are linear superpositions of the ground and the charge transfer states of the DAC7

10.1021/jp1072796  2010 American Chemical Society Published on Web 11/23/2010

Photoexcitation of Donor-Acceptor Complexes ( - + Ψ( ) c( 1 Ψ(AD) + c2 Ψ(A D )

J. Phys. Chem. A, Vol. 114, No. 50, 2010 12999

(1.1)

where the diabatic states Ψ(AD) and Ψ(A-D+) are pure ground and pure radical-ion pair states, accordingly. The ionicity of the upper adiabatic state is equal to |c2+|2. Immediately after optical excitation, the system is far from the crossing point of the diabatic terms, so that the energy gap between them is much larger than the electronic coupling, and the degree of excited state ionicity is close to one. During relaxation, the system moves in the direction of the upper adiabatic term minimum and approaches the intersection of the diabatic terms, where the ionicity degree is close to 0.5. The experimental data, however, indicate that the ionicity of excited DACs increases rather than decreases with time.12 The main limitation of the two approaches last mentioned is full neglect of the reorganization of intramolecular highfrequency modes in contrast to the mechanism proposed in ref 2. Most probably the truth lies somewhere in between. Indeed, the reorganization of intramolecular modes can strongly increase the rate constant of both highly exergonic electron transfer reactions13-17 and weakly exergonic nonequilibrium CR.17-19 Moreover, for quantitative description of the CR dynamics, the real spectrum of high-frequency modes should be employed.20,21 Such a spectrum typically includes 5-10 active modes. The decay of excited vibrational states due to intramolecular vibrational redistribution and relaxation can also significantly affect the CR dynamics.19 A model involving the reorganization of both the solvent and intramolecular modes was used for explanation of the drivingforce dependence of back electron transfer in contact radicalion pairs in ref 22. The shallow, nearly linear, free energy dependence observed with a series of DACs in weakly polar solvent was elucidated in terms of an increase of the solvent reorganization energy with decreasing driving force. The validation of this trend was obtained from fluorescence spectra analysis. The increase of the solvent reorganization energy for this series of DACs22 is expected since it correlates with decreasing number of methyl substituents on the donor, that is, with decreasing the donor size. This conclusion directly appears from Marcus expression for the reorganization energy of a polar solvent. The aim of this paper is: (i) to fit the available experimental data on the free energy gap law to the generalized stochastic point-transition model, (ii) to investigate the extent of the vibrational nonequilibrium in the CR of excited DACs, and (iii) to ascertain the extent of deviation of CR dynamics from exponential decay.

diffusion time, we also neglect the alteration of the parameters due to mutual motion of the donor and acceptor. To quantitatively describe the ultrafast CR in the excited complexes we use well proven stochastic point-transition approach,5,6 which could allow simulating the CR kinetics beyond the domain of the golden rule applicability when the nonthermal electron transfer probability is significant. Recently, the stochastic point-transition approach was generalized to multilevel systems.17,18,21 The multichannel stochastic model developed takes into account:18,21 (1) reorganization of the solvent as well as the intramolecular high-frequency vibrational modes of DAC; (2) the relaxation of intramolecular vibrational excited states; (3) the formation of the electronic excited state with a strongly nonequilibrium nuclear configuration by a short laser pulse. The model allows simulating the effect of the spectral characteristics of the pump pulse (the carrier frequency of excitation pulse, ωe, and its duration, τe) on CR dynamics in the excited DACs. The adequacy of this model was tested for a number of DACs and solvents.21,23,24 Within this model a temporal evolution of the system “DAC+solvent” can be described by a set of differential equations for the probability distribution functions for the electron-vibrational states considered. Further, we assume that only two electronic states of the complex can be populated in the CR processsthe neutral ground state |g〉 and the excited ionic (charge separated) state |e〉. In this case the set of equations for the probability distribution functions for the electronic b) excited state, Fe(Q1, Q2, t), and the ground neutral state, F(n g (Q1, 21 Q2, t), can be written as

∂Fe ) LêFe ∂t

∑

b) kbn (Q1, Q2)(Fe - F(n g )

(2.1)

n1,n2,...,nM

b) ∂F(n g b) (n b) ) LˆgF(n g - kb n (Q1, Q2)(Fg - Fe) + ∂t 1 F(nbR′ ) (nR+1) g τ R VR

∑

∑ τ(n1 ) F(ngb) R

R

(2.2)

VR

where the Smoluchowski operators Lˆg and Lê 2

Lˆg )

(

2

∂ ∑ τ1i 1 + Qi ∂Q∂ i + 〈Qi2〉 ∂Q 2 i)1

i

)

(2.3)

II. The Multichannel Stochastic Model Donor-acceptor complexes are fairly labile systems and can exist in different geometric configurations. Naturally, different sets of charge transfer parameters (reaction free energy, solvent reorganization energy, electronic coupling) associate with each configuration. Considering that (i) complexes largely absorb radiation in configurations with the largest electronic matrix element of transition to the ionic state and (ii) the matrix element is sensitive to the complex geometry, much fewer diversity of the configurations of the complexes excited by a pulse with narrow spectrum (the width of the pulse spectrum is much fewer than the width of the charge transfer band) is expected. As first approximation we accept that all excited DACs have the same charge transfer parameters. Taking into consideration that ultrafast CR proceeds on a time scale much shorter than the

2

Lê )

(

2

∂ ∑ τ1i 1 + (Qi - 2Eri) ∂Q∂ i + 〈Qi2〉 ∂Q 2 i)1

i

)

(2.4)

describe the diffusion on the diabatic free energy surfaces of the electronic states Ug(nb) and Ue

b) U(n g

Q21 Q22 ) + + 4Er1 4Er2

∑ nRpΩR + ∆GCR R

(2.5)

13000

J. Phys. Chem. A, Vol. 114, No. 50, 2010

Ue )

(Q1 - 2Er1)2 4Er1

+

(Q2 - 2Er2)2 4Er2

Yudanov et al.

(2.6)

Here, ∆GCR is the CR free energy, Q1 and Q2 are the reaction coordinates corresponding to different solvent modes with the solvent reorganization energy Er1 and Er2.25 In eqs 2.1-2.2 Fg(nb)(Q1, Q2, t) is the probability distribution function for the ground neutral state with the excitation of nR (nR ) 0, 1, 2,...) vibrational quanta for Rth intramolecular modes with the n has Mfrequency ΩR (R ) 0, 1, 2,..., M). The vector b components (n1, n2,..., nR,..., nM). The vector b nR′ differs from b n only by the number of vibrational quanta for Rth mode b nR′ ) (n1, n2,..., nR + 1,..., nM). So, the model accounts for the reorganization of a number of intramolecular high-frequency vibrational modes that generally leads to the vibrational sublevels of both the ground and excited states (Figure 1). However, vibrational sublevels of the excited state may be omitted from consideration owing to their fast relaxation. Figure 1 shows several vibrationally excited sublevels of the ground state and also intersection points of cuts of the free energy surfaces in the case M ) 2. In the Appendix the set of eqs 2.1-2.2 for this case is written in more detail. We adopt here a single-quantum mechanism of highfrequency mode relaxation and the transitions nR f nR - 1 (nR) . Naturally, the ground proceed with the rate constant 1/τVR (0) ) ∞ must be vibrational state is stable and the equation τVR fulfilled. In particular, this condition is met for the model (nR) (1) ) τVR /nR. τVR In eqs 2.3-2.4 the following notations are used: 〈Q2i 〉 ) 2ErikBT is the dispersion of the equilibrium distribution along the ith reaction coordinate (i ) 1, 2), kB is the Boltzmann constant, and T is the temperature. Electron transitions between the excited state |e〉 and a vibrational sublevel of the ground state |g〉 are described by the parameters

2πVb2n 2πVb2n b) kbn ) U ) ) δ(U(n δ(z - zb†n ), g e p p Vb2n ) Vel2Fbn , Fbn )

∏ R

SnRRe-SR nR !

(2.7)

where z ) ΣQi is the collective energetic reaction coordinate, zbn† ) Erm - ∆GCR - ΣRnRpΩR are the points of the intersection of terms Ug(nb) and Ue, Erm is the solvent reorganization energy. Fbn is the Franck-Condon factor, and SR ) ErVR/pΩR and ErVR are the Huang-Rhys factor and the reorganization energy of the Rth high-frequency vibrational mode, respectively. Here we use a definition of the CR reaction coordinate as a difference between the corresponding energy levels ∆E(t) ) b) † U(n g - Ue ) z - zb n that allows describing the solvent relaxation through the autocorrelation function K(t) ) 〈∆E(t)∆E(0)〉.5 In this equation zbn† is a constant determining the coordinate origin. For the Debye model of solvent relaxation, this function is written in the form5

K(t) ) 2ErmkBTX(t)

where X(t) ) e-t/τL, and τL is the longitudinal dielectric relaxation time. The exponential form of K(t) implies that the relaxation process is Markovian. In this case, the motion along the reaction coordinate is diffusive. The real solvents such as acetonitrile (ACN) are characterized by at least two relaxation time scales.26-28 These time scales are attributed to different relaxation modes corresponding to distinct types of motions of the solvent molecules. If the relaxation modes are independent, as usually assumed, then the autocorrelation function K(t) is a sum of contributions from each mode. Using the Markovian approximation for all solvent modes, one can write K(t) in the same form as in eq 2.8 but with the solvent relaxation function, X(t), as a sum of exponentials25,29 N

X(t) )

∑ xie-t/τ

(2.9)

i

1

Here xi ) Eri/Erm, τi are the weight and the relaxation time constant of the ith medium mode, respectively; Erm ) Σ1NEri, and N is the number of the solvent modes. Such solvents can be described in terms of N diffusion coordinates Qi.25 Although in real polar solvents, the exponential stage of the autocorrelation function decay is only achieved at relatively long times and a considerable part of decay is inertial,26-28,30 eq 2.9 is applicable to description of the real solvents. This possibility is discussed in detail in ref 21. Henceforward, we shall approximate the solvent relaxation function with eq 2.9 considering N ) 2. All simulations are performed at room temperature, T ) 300 K, and the dynamic parameters of the solvent (ACN) are equal to τ1 ) 0.19 ps, τ2 ) 0.50 ps, x1 ) x2 ) 0.5.8 To specify the initial conditions, we assume that the system is initially in the ground state with the nuclear coordinates distributed according to the Boltzmann law. Assuming that the pump pulse is short enough, one can obtain the following general expression for the initial probability distribution function on the excited term31

Fe(Q1, Q2, t ) 0) ) Z-1 Figure 1. The multichannel CR to the ground state of DAC resulting in the excitation of a number of sublevels of the high-frequency intramolecular vibrational modes (dashed lines). The vertical arrows stand for vibrational relaxation. The initial excited state distribution produced by a laser pump is also pictured.

(2.8)

{

exp -

nR

b) (pδω(n e -

[

∑ ∏ R

∑ Q˜i)2τ2e

2p2

]

SnRRe-SR × nR ! -

∑

˜ i2 Q 4ErikBT

}

(2.10)

Photoexcitation of Donor-Acceptor Complexes


TABLE I: Parameters of High-Frequency Vibrational Modes for CR in a Series of Excited DACs IDU-TCNEa

PhCP-TCNEb

R

pΩR, eV

SR

pΩR, eV

SR

1 2 3 4 5 6 7 8 9 10

0.0558 0.0674 0.0712 0.0767 0.1195 0.1428 0.1608 0.1932 0.1996 0.2762

0.0309 0.0464 0.0464 0.0155 0.0232 0.0309 0.1237 0.8813 0.1546 0.5411

0.1272 0.1469 0.1823 0.1935 0.1993

0.1245 0.1211 0.1143 0.5150 0.2302

a

Reference 21. b Reference 35.

b) ˜ i ) Qi - 2Er , pδω(n where Q e ) pωe + ∆GCR - Erm - ∑RnRpΩR, i and Z is a factor specifying the initial fraction of the excited DACs. Its value is irrelevant for the rate constant calculations and thus is not stated. The system of eqs 2.1-2.2 with the initial condition eq 2.10 is solved numerically using the Brownian simulation method.31,32 The normalized population of the excited state is given by eq 2.11

Pe(t) )

∫∫ Fe(Q1, Q2, t)dQ1 dQ2 [∫∫ Fe(Q1, Q2, t ) 0)dQ1 dQ2]-1

Figure 2. The free energy dependence of the CR rate constant, kCR in s-1, for a number of electronic coupling values in the excited DACs: 1, Naph-TCNE; 2, Py-TCNE; 3, Per-TCNE; 4, Naph-TCNQ; 5, PyTCNQ; 6, Naph-PMDA; 7, Chr-PMDA; 8, Py-PMDA; 9, Per-PMDA; 10, Py-PA; 11, An-PA; 12, Per-PA; 13, HMB-Methylbenzene; 14, PMB-Methylbenzene; 15, DUR-Methylbenzene; 16, TMB-Methylbenzene; and 17, p-XY-Methylbenzene. The experimental data and notations were taken from refs 2 and 22. Dash-dot straight line is a linear approximation to the experimental data.2 For comparison the dependencies of the thermal rate constant eq 3.1 are shown as dashed curves. The high-frequency vibrational spectrum involving 5 modes is employed in simulations (see Table I). Parameters used: Vel ) 0.02 eV, Erm ) 0.5 eV (lines a, c); Vel ) 0.0065 eV, Erm ) 0.6 eV (lines b, d); ErV ) 0.51 eV. The experimental rate constants of CR were measured in ACN solution (complexes 1-12)2 and (13-17)22 in chloroform solution.

(2.11)

and the effective rate constant of CR is determined by eq 2.12 -1 kCR )

∫0∞ Pe(t)dt

(2.12)

The model accounts for the local reversibility of electron transfer that can be adequately described only if the intramolecular vibrational relaxation or the vibrational redistribution is also taken into consideration. The intramolecular vibrational redistribution is well-known to proceed on the time scale of τV (1) ∼ 100 fs.33 In simulations the value τVR ) 150 fs is set for all 34 vibrational modes. The spectral densities of high-frequency vibrational modes for the complexes considered are not known. A possible solution of this problem was suggested in ref 23. For a fixed number of high-frequency vibrational modes, the CR rate was shown to depend weakly on the vibrational spectral density provided the total reorganization energy is constant. It implies that a universal spectral density including 5-10 high-frequency modes with variable total reorganization energy may be exploited as a good approximation for any DAC. As universal spectral density we accept high-frequency vibrational spectra of DACs consisting of phenylcyclopropane (PhCP), isodurene (IDU) as electron donors and tetracyanoethylene (TCNE) as electron acceptor.23,35 The spectra of these complexes involve 5 and 10 active vibrational modes. The values of parameters are given in Table I. III. Simulation Results and Discussion Free Energy Gap Dependence. The results of numerical simulation of the CR dynamics for a number of DACs are presented in Figures 2 and 3. The experimental data borrowed from refs 2, 8, and 36 are pictured by symbols. For comparison the thermal CR rate constant determined by eq 3.1

Figure 3. The free energy dependence of the CR rate constant in the excited DACs: 1, IDU-TCNE; 2, PMB-TCNE; 3, HMB-TCNE; 4, TMB-TCNE; 5, ANI-PDMA; 6, VER-PDMA; 7, DMB-PDMA; 8, TrMB-PDMA; and 9, TeMB-PDMA. The experimental data and notations were taken from refs 8 and 36. Dashed curves (lines c, d) are the thermal rate constant. Parameters are: Vel ) 0.12 eV (lines a, c); Vel ) 0.07 eV (lines b, d); Erm ) 0.5 eV, ErV ) 0.51 eV.

kth )

Vel2 p

π ErmkBT

[

∑ ∏

[

nR

exp -

R

]

e-SRSnRR × nR !

(∆GCR + Erm + ∑nRpΩR)2 R

4ErmkBT

]

(3.1)

is displayed by dashed lines. This equation is straightforward generalization of well-known expression.13 At first we tried to reproduce the free energy dependence of the CR rate constant on the presumption that the parameters Vel, Erm, and ErV are free but invariable through the series of DACs. With this assumption the model elaborated fails to reproduce the linear dependence of the logarithm of CR rate constant throughout the experimentally accessible region of the

13002


reaction free energies -3 eV < ∆GCR < -0.5 eV (solid straight line in Figure 2). Nearly linear dependence of log kCR on ∆GCR with the same slope (-1.35 decade/eV) can be obtained for larger values of electronic coupling (see Figure 3), however such curves lie at least one decade higher than the experimental line. A way out may be found in experimental data. Comparing the data for DACs with numbers 2 and 5 or 1 and 4 in Figure 2, one can see that the difference in CR rate constants of the DACs with very close values of -∆GCR can be as large as ≈10. This difference directly testifies that there is a considerable variation of other DAC parameters in the series. In current literature there are data showing some variation of the parameters Erm and ErV with DACs. In ref 22 an increase of the medium reorganization energy, Erm, with the rise of the driving force was obtained for a series of DACs consisting of methyl-substituted benzenes and 1,2,4,5-tetracyanobenzene from fluorescence spectra analysis. On the other hand, for similar DACs consisting of methyl-substituted benzenes and TCNE, the opposite trend was revealed using the analysis of both the stationary charge transfer band and resonance Raman spectra.21 For this reason we do not expect a global trend in the solvent reorganization energy variation on the region of the reaction free energies -3 eV < ∆GCR < -0.5 eV. For the DACs considered, a correlation between the medium reorganization energy and CR free energy is ignored. Of course, there is some dispersion of the solvent reorganization energy magnitudes that may worsen the fitting quality (see points 13-17 in Figure 2). The problem of the reorganization energy determination could be resolved if the experimental data on stationary charge transfer absorption bands and resonance Raman spectra for all DACs under study were available.20,21,37,38 A good fit can be obtained if we separate the DACs into two groups with different values of electronic coupling. By this means we can reproduced the dependence kCR(-∆GCR) for each group separately (solid lines a and b in Figure 2). Here, a value of Erm ) 0.5 eV was used. This value is close to Erm ) 0.48 eV, which was previously determined for contact radical-ion pairs in acetonitrile.39 The best fit to experimental data2 is obtained for the electronic coupling values Vel ) 0.02 eV (line a) and Vel ) 0.0065 eV (line b). The DACs investigated in refs 8 and 36 revealed considerably faster CR dynamics (see Figure 3), and these data can be fitted with larger values of electronic coupling: Vel ) 0.12 eV (line a) and Vel ) 0.07 eV (line b). However, the stochastic approach still applicable.21 Numerical simulations have shown that the CR rate constant decreases monotonically with increasing CR exothermicity throughout the accessible range of the reaction free energies, - 3 eV < ∆GCR < -0.5 eV, if electronic coupling Vel > 0.02 eV for typical magnitudes of the reorganization energy. For smaller values of electronic coupling the Marcus normal region arises (line b in Figure 2). In this case the dependence of the effective rate constant on the reaction exothermicity has, typical for thermal reactions, a bell-shaped form with a small deviation from the thermal rate in the vicinity of the maximum. The group of complexes with small electronic coupling is characterized by rather low values of CR rate constant, significantly lower than the solvent relaxation rate. This is the reason why the relaxation of nonequilibrium nuclear state produced by the photoexcitation results in minor difference between effective and thermal rate constants. Nonequilibrium Effects on CR Dynamics. Figures 2 and 3 demonstrate considerable difference between effective and thermal rates. One can distinguish three regions of exothermicity. A strongly exothermic region where the effective CR rate is

Yudanov et al.

Figure 4. The coarse grain probability distributions of electronic transitions, Y(z), over the reaction coordinate, z, for the nonthermal (panels a, c) and the thermal (panels b, d) recombination. The panels a and b correspond to the Per-TCNE complex (∆GCR ) -0.56 eV), and the panels c and d to Py-TCNE (∆GCR ) -0.91 eV) in ACN. Parameters used: Vel ) 0.02 eV, Erm ) 0.5 eV, ErV ) 0.51 eV.

close to the thermal one. In the region of moderate exothermicity kCR is less than kth and in the weakly exothermic region the inverse relation, kCR > kth, is fulfilled. The deviation of the effective CR rate constant from its thermal value is associated with the nonequilibrium nature of the CR. For weakly exothermic reactions (-∆GCR < Erm) the wave packet passes through the most powerful sinks during the solvent relaxation. As a result, the CR rate at the hot stage exceeds the thermal one. The coarse grain probability distributions of electronic transitions, Y(z), over the reaction coordinate z are shown in Figure 4a for the complex Per-TCNE in ACN solution (-∆GCR ) 0.56 eV, Vel ) 0.02 eV). This quantity is determined by eq 3.2

Y(z) )

∑ ∫ dt ∫ dQ1 ∫ dQ2kbn(Q1, Q2)[Fe(Q1, Q2, t) -

n∈∆z

b) F(n g (Q1, Q2, t)]

(3.2)

Here the sum runs all sinks located on an interval from z ∆z to z, and ∆z ) kBT. For comparison, the probability distribution is also pictured for the thermal process (Figure 4b). As seen for the thermal recombination, the most effective sinks are located in the vicinity of the Ue term minimum (z ) 2Erm), and for the nonthermal process they are scattered in a wide region of lesser values of z. The huge difference between these distributions is an evidence of crucial role of the hot transitions. In spite of the fact that the effective and thermal rate constants at -∆GCR ) 0.56 eV have practically the same value, the CR in the complex Per-TCNE in ACN solution mainly proceeds in nonthermal regime. In the area of lesser values of -∆GCR the thermal rate decreases because of the Marcus activation barrier and the effective rate constant can even rise since the wave packet is initially placed in the area of more powerful sinks so that the inequality kCR > kth is held. In the area of moderate exothermicity the most powerful sinks are located in the neighborhood of the excited state term minimum. The wave packet created by pump pulse takes some time (of the order of the medium relaxation time) to reach the most powerful sinks. This implies that the CR rate constant increases with time approaching its maximum value, kth, and hence kCR < kth. For nonthermal recombination (Figure 4c) the active sinks cover a rather wide region 0.5 e z/2Erm e 2.25, whereas for the thermal process this region is narrower by a

Photoexcitation of Donor-Acceptor Complexes


factor of approximately two (Figure 4d). This difference clearly shows that in the area of moderate exothermicity an essential part of DACs recombines at the hot stage. In the area of strong exothermicity, -∆GCR > 2.25 eV, the wave packet during the medium relaxation intersects with only weak sinks so that the hot transitions becomes irrelevant. Obviously, in the beginning the rate constant rises with time (nonstationary phase), but the time scale of this phase is much shorter than the reaction time so that the reaction predominantly proceeds in the thermal regime and the effective rate approaches the thermal one, kCR = kth. The nonequilibrium effects fade away with a decrease of electronic coupling (see the bottom curve in Figure 2) and are enhanced for larger values of electronic coupling (see Figure 3). Nonexponentiality of CR in Excited DACs. The thermal CR is well-known to proceed in the exponential regime. For fast and especially ultrafast CR, the nuclear subsystem can be far from its equilibrium due to the preparation of the initial state. Besides, a fast reaction itself can perturb the nuclear equilibrium. So the nuclear nonequilibrium should be considered as a regular event in ultrafast CR. The experimental investigations of the CR with time resolution of 30-40 fs uncovered substantial deviations of the ultrafast CR dynamics from exponential.36 The decay of the DAC excited-state population, Pe(t), can be fitted to the function

Pe(t) ) exp{-(t/τ)s}

(3.3)

where s is a free parameter. A good fit was obtained for a series of DACs (IDU-TCNE, PMB-TCNE, and HMB-TCNE) pictured in Figure 3, points 1, 2, and 3 with s varying in the interval from 1.1 to 1.6.36 The model considered reproduces observed nonexpnentiality.21 The dependence of the parameter s on the excitation pulse carrier frequency observed in experiments with high time resolution36 is also evidence of nonequilibrium nature of the excited DACs recombination. Obviously, the contrary assertion is not true, that is, the exponential decay of the excited complexes is not direct evidence of the equilibrium regime of CR. For example, the CR in the excited complex Per-TCNE in acetonitrile is known to proceed in exponential regime.2 This reaction is weakly exothermic and occurs mainly in the nonthermal regime (compare Figures 4a and 4b). The simulations of the excited state population decay for few models with different spectra of high-frequency vibrational modes are pictured in Figure 5. The figure shows that in the absence of quantum mode reorganization (curve 0) the decay is highly nonexponential. Even the model with single quantum mode (ErV ) 0.51 eV and pΩ ) 0.17 eV) approaches the decay to exponential with s ) 0.94 with rather high quality of the fit, R2 ) 0.9964 (curve 1). For real vibrational spectra involving 5 and 10 modes the CR dynamics approximate still further to exponential decay. The parameter of nonexponentiality is: s ) 1.09 for M ) 5, R2 ) 0.9973 (curve 5), and s ) 1.13 for M ) 10, R2 ) 0.9982 (curve 10). These theoretical results correspond closely to experimental data. In Figure 6 the time dependence of the measured2 and simulated signals A(t) are pictured. To reproduce the signal measured in experiments, A(t), the excited state population given by eq 2.11 should be convoluted with the instrument response function

Figure 5. The time dependence of the excited-state population dynamics. The parameters are: ∆GCR ) -0.5 eV, Vel ) 0.02 eV, Erm ) 0.5 eV, and ErV ) 0.51 eV. The indexes near the lines indicate the number of active intramolecular high-frequency modes of the DAC (see Table I).

Figure 6. The time-dependent signal A(t) for the Per-TCNE complex in ACN. Results of numerical simulation are shown as a solid line and experimental data as circles.2 Parameters used are as in Figure 4 for the Per-TCNE complex, τ0 ) 0.57 ps.

A(t) ) (πτ20)-1/2

∫-∞0 Pe(t - t1)exp(-t12/τ20) dt1 (3.4)

with τ0 ) 0.57 ps. It should be noted that CR dynamics in this case is too fast compared with experimental resolution. As a result some nonexponentiality of the dynamics could be missed. In the area of smaller magnitudes of -∆GCR < Erm the activation barrier dividing reactant and product state minima becomes significant and two-step CR dynamics is expected.40 The time scales of faster hot stage and slower thermal one can strongly differ. Such a behavior was observed experimentally for complexes consisting of 1,2,4-trimethoxybenzene (TMB) and TCNE36 (point 4 in Figure 3). The recombination dynamics of the photoexcited complex in ACN revealed its two-stage character. The first stage was very fast with effective time of 30-80 fs, and the second, slow with effective time longer than 100 ps. In more viscous solvents, the reaction terminated at the fast stage. These regularities are well described by the considered here model that is a strong argument in favor of the nonadiabatic model.24 Two-step CR dynamics is unique so far. There are at least two reasons for its rarity. First, in the area of low exothermicity, -∆GCR < 0.5 eV, exciplexes can dominate.12,41 The exciplexes are characterized by strong electronic coupling and by quite different regularities. Second, the CR can terminate at the hot stage that results in unobservability of the second thermal stage.24 IV. Concluding Remarks In this paper we have illustrated the efficiency of multichannel stochastic model for quantitative description of the ultrafast CR in a number of DACs in the nonadiabatic regime.

13004


Yudanov et al.

The main conclusions can be summarized as follows. (1) The model predicts monotonic decrease of the CR rate constant with the reaction exothermicity growth for the DACs with not too weak electronic coupling. Multichannel hot transitions accelerating the CR in the low exergonic region overturn the ascending branch of the Marcus inverted parabola. Moreover, the nonequilibrium effect suppresses the CR rate in the region of moderate exothermicity. As a result, the dependence of the CR rate constant on the free energy gap is straightened in full accord with experimental data. (2) The presence of DACs with close values of -∆GCR but with the CR rate constant varying by the factor of 10 or more (compare the data for DACs with numbers 2 and 5 or 1 and 4 in Figure 2 and those for DACs with -∆GCR in interval from -1.3 to -2.3 in Figures 2 and 3) strongly evidence that the other parameters alter within the collection of DACs. It appears that the free energy dependence of the CR rate constant can be well fitted to the current theory if the DACs are separated into a few groups with different values of the electronic coupling. Good fit testifies that other parameters within the collection of the DACs are weakly variable. (3) The model provides correct description of the nonexponential CR dynamics in excited DACs. The two-step CR predicted by the model has been observed in TMB-TCNE complex in ACN. (4) The model correctly describes the influence of the solvent relaxation time scales on the CR dynamics, in particular, an intensification of CR with increasing the solvent relaxation time and disappearance of the thermal CR stage in more viscous solvents. Acknowledgment. We are indebted to Dr S. Feskov and Mr. V. Ionkin for the usage of their software. This work was supported by the Ministry of education and science of the Russian Federation (contract P1145) and the Russian foundation for basic research (Grant No. 10-03-97007). Appendix Explicit Form of Evolution Equations of the Generalized Stochastic Model. In this appendix the set of eqs 2.1-2.2 for the probability distribution functions Fe(Q1, Q2, t) and Fg(nb)(Q1, Q2, t) are written in explicit form for only two intramolecular high-frequency modes

∂Fe ) LêFe ∂t

[

∑ kn ,n (Q1, Q2)(Fe - F(ng ,n )) 1

n1,n2

1

2

2

(A1)

1,n2) ∂F(n g 1,n2) 1,n2) - kn1,n2(Q1, Q2)(F(n - Fe) + ) LˆgF(n g g ∂t 1 (n1+1,n2) 1 1 1 1,n2+1) 1,n2) F + (n +1) F(n - (n ) + (n ) F(n g g (n1+1) g 2 1 τV1 τV2 τV1 τV22 (A2)

] [

]

The scheme of transitions corresponding to these equations is pictured in Figure 1. Only a part of the transitions is drawn.

References and Notes (1) Asahi, T.; Mataga, N. J. Phys. Chem. 1989, 93, 6575. (2) Asahi, T.; Mataga, N. J. Phys. Chem. 1991, 95, 1956. (3) Marcus, R. A. J. Chem. Phys. 1956, 24, 966. (4) Tachiya, M.; Murata, S. J. Am. Chem. Soc. 1994, 116, 2434. (5) Zusman, L. D. Chem. Phys. 1980, 49, 295. (6) Yakobson, B. I.; Burshtein, A. I. Chem. Phys. 1980, 49, 385. (7) Frantsuzov, P. A.; Tachiya, M. J. Chem. Phys. 2000, 112, 4216. (8) Nicolet, O.; Vauthey, E. J. Phys. Chem. A 2002, 106, 5553. (9) Hubig, S. M.; Bockman, T. M.; Kochi, J. K. J. Am. Chem. Soc. 1996, 118, 3842. (10) Segawa, H.; Takehara, C.; Honda, K.; Shimidzu, T.; Asahi, T.; Mataga, N. J. Phys. Chem. 1992, 99, 503. (11) Mikhailova, V. A.; Ivanov, A. I.; Vauthey, E. J. Chem. Phys. 2004, 121, 6463. (12) Mataga, N.; Miyasaka, H. AdV. Chem. Phys. 1999, 107, 431. (13) Jortner, J.; Bixon, M. J. Chem. Phys. 1988, 88, 167. (14) Walker, G. C.; Akesson, E.; Johnson, A. E.; Levinger, N. E.; Barbara, P. F. J. Phys. Chem. 1992, 96, 3728. (15) Barbara, P. F.; Walker, G. C.; Smith, T. P. Science 1992, 256, 975. (16) Denny, R. A.; Bagchi, B.; Barbara, P. F. J. Chem. Phys. 2001, 115, 6058. (17) Bagchi, B.; Gayathry, N. AdV. Chem. Phys. 1999, 107, 1. (18) Feskov, S. V.; Ionkin, V. N.; Ivanov, A. I. J. Phys. Chem. A 2006, 110, 11919. (19) Mikhailova, V. A.; Ivanov, A. I. J. Phys. Chem. C 2007, 111, 4445. (20) Wynne, K.; Galli, C.; Hochstrasser, R. M. J. Chem. Phys. 1994, 100, 4797. (21) Feskov, S. V.; Ionkin, V. N.; Ivanov, A. I.; Hagemann, H.; Vauthey, E. J. Phys. Chem. A 2008, 112, 594. (22) Gould, I. R.; Noukakis, D.; Gomez-Jahn, L.; Goodman, J. L.; Farid, S. J. Am. Chem. Soc. 1993, 115, 4405. (23) Ionkin, V. N.; Ivanov, A. I. Chem. Phys. 2009, 360, 137. (24) Ionkin, V. N.; Ivanov, A. I.; Vauthey, E. Rus. J. Phys. Chem. A 2009, 83, 683. (25) Zusman, L. D. Chem. Phys. 1988, 119, 51. (26) Rosenthal, S. J.; Xie, X.; Du, M.; Fleming, G. R. J. Chem. Phys. 1991, 95, 4715. (27) Maroncelli, M.; Kumar, V. P.; Papazyan, A. J. Phys. Chem. 1993, 97, 13. (28) Jimenez, R.; Fleming, G. R.; Kumar, P. V.; Maroncelli, M. Nature 1994, 369, 471. (29) Garg, S.; Smith, C. J. Phys. Chem. 1965, 69, 1294. (30) Gumy, J. C.; Nicolet, O.; Vauthey, E. J. Phys. Chem. A 1999, 103, 10737. (31) Fedunov, R. G.; Feskov, S. V.; Ivanov, A. I.; Nicolet, O.; Page`s, S.; Vauthey, E. J. Chem. Phys. 2004, 121, 3643. (32) Gladkikh, V.; Burshtein, A. I.; Feskov, S. V.; Ivanov, A. I.; Vauthey, E. J. Chem. Phys. 2005, 123, 244510. (33) Kovalenko, S. A.; Schanz, R.; Hennig, H.; Ernsting, N. P. J. Chem. Phys. 2001, 115, 3256. (34) Elsaesser, T.; Kaiser, W. Annu. ReV. Phys. Chem. 1991, 42, 83. (35) Myers Kelly, A. J. Phys. Chem. A 1999, 103, 6891. (36) Nicolet, O.; Banerji, N.; Page`s, S.; Vauthey, E. J. Phys. Chem. A 2005, 109, 8236. (37) Markel, F.; Ferris, N. S.; Gould, I. R.; Myers, A. B. J. Am. Chem. Soc. 1992, 114, 6208. (38) Kulinowski, K.; Gould, I. R.; Myers, A. B. J. Phys. Chem. 1995, 99, 9017. (39) Gould, I. R.; Young, R. H.; Farid, S. J. Phys. Chem. 1991, 95, 2068. (40) Ivanov, A. I.; Potovoi, V. V. Chem. Phys. 1999, 247, 245. (41) Gould, I. R.; Farid, S. J. Phys. Chem. B 2007, 111, 6782.

JP1072796

Nonequilibrium Phenomena in Charge Recombination of Excited

Recommend Documents