Kinetics of Nonequilibrium Electron Transfer in Photoexcited

ARTICLE pubs.acs.org/JPCA

Kinetics of Nonequilibrium Electron Transfer in Photoexcited Ruthenium(II)-Cobalt(III) Complexes Serguei V. Feskov, Anna O. Kichigina, and Anatoly I. Ivanov* Volgograd State University, University Avenue 100, Volgograd, 400062, Russia ABSTRACT: Kinetics of photoinduced electron transfer reactions in [Ru II (L-L)Co III ]5þ complexes have been investigated in the framework of the stochastic point-transition model. The model involves the medium and intramolecular nuclear reorganization as well as fast relaxation of intramolecular high frequency vibrations and description of the medium relaxation in terms of two time scales. The model has allowed reproducing the experimental data (Torieda, H.; Nozaki, K.; Yoshimura, A.; Ohno, T. J. Phys. Chem. A 2004, 108, 4819) of forward and backward electron transfer kinetics, including the low yield of electron transfer products and its variation with solvent. These results have lent support to the important role of hot backward electron transfer in the formation of low yield of electron transfer products. The experimentally observed significant decrease of the product yield in more viscous solvents has been shown to be a direct consequence of the hot transition efficiency increase. A weak opposite dependence, also revealed in experiments, has been elucidated in terms of two time scales of the solvent relaxation. Solvent relaxation is well-known to involve at least two stages: the inertial one (fast) and the diffusive one (slower) with the time scales weakly dependent on and proportional to solvent viscosity, respectively. When hot transitions are terminated at the stage of the inertial relaxation, the yield of the electron transfer products is nearly independent of the diffusive time scale; otherwise a strong increase of the yield in more viscous solvents should be observed.

I. INTRODUCTION Photoinduced charge transfer (CT) in condensed media is often followed by return CT (RCT) to the ground state reducing the quantum yield of the CT product formation. Apparently, the low product yield can be observed if the thermal rate of recombination, kRCT, is much larger than the rate of charge separation, kCT. For example, this is possible when the activation barrier of forward CT is higher than that of RCT (Figure 1A). However, a similar result can also be observed in the opposite limit kRCT , kCT when the products are long-lived. This occurs if the relation between the activation barrier heights is inverted (Figure 1B). The reduction of the CT product yield in this case is due to the so-called “hot” transitions proceeding in parallel with nuclear relaxation, and kCT and kRCT are thought of as the equilibrium (thermal) rate constants. It has long been known that hot or nonequilibrium CT can be of primary importance in many chemical systems and has been actively studied for the past two decades.1-12 The disposition of terms pictured in Figure 1B is typical for electron transfer from the second excited state. In experiments on derivatives of porphyrin the hot transitions were shown to play a key role in such a transfer.13-18 Measurements of the ion state concentration carried out in a recent paper19 verified this conclusion directly. A scheme of electronic terms similar to that in Figure 1B is realized in ruthenium(II)-cobalt(III) charge transfer compounds investigated experimentally in a series of papers.20-22 r 2011 American Chemical Society

In these papers photoinduced electron transfer reactions of [(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ (tpy = 2,20 :60 ,200 -terpyridine and tpy-ph-tpy = 1,4-bis[2,2 0 :6 0 ,2 00 -terpyridine-4 0 yl]benzene) and [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ (tpy-tpy = 60 ,600 -bis(2-pyridyl)-2,20 :40 ,400 :200 ,2000 -quarterpyridyne) were studied by means of subpicosecond transient absorption spectroscopy. Molecular structures of these compounds are shown in Figure 2. In the experiments the kinetics of the excited state population decay and the ground state population recovery were detected. The yield of the relaxed electron transfer products was found to be rather low (down to 0.2) although the return electron transfer to the ground state proceeds considerably more slowly than the forward one.22 This allowed the authors to suggest that the reduction of the yield is mainly due to backward transition of nonrelaxed electron transfer products to the original reactant state at the hot stage. It was also supposed that the decay of the electron transfer product state leads to population of both the reactant ground state [(tpy)1RuII(L-L)1CoIII(tpy)]5þ and the triplet excited state [(tpy)1RuII(L-L)3CoIII(tpy)]5þ (L-L = tpy-ph-tpy and tpy-tpy). There is one more argument in favor of the important role of hot transitions in RuII-CoIII compounds. The theory predicts a larger hot transition efficiency in slower solvents.9,12 Precisely Received: September 9, 2010 Revised: December 3, 2010 Published: February 10, 2011 1462

dx.doi.org/10.1021/jp108607t | J. Phys. Chem. A 2011, 115, 1462–1471

The Journal of Physical Chemistry A

Figure 1. Diabatic electronic terms involved in photoinduced charge transfer (CT) and the ensuing return charge transfer (RCT) processes: (A) weakly exergonic CT and highly exergonic RCT; (B) highly exergonic CT and weakly exergonic RCT.

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Figure 3. Scheme of electronic states and photochemical processes in [(tpy)RuII(L-L)CoIII(tpy)]5þ (L-L = tpy-tpy, tpy-ph-tpy) complexes. Abbreviations EET, HT, and RET are used to indicate the excited state electron transfer, hole transfer, and return electron transfer processes.

inverted region.8,23-25 However, what is more important for the system considered is that these modes can also significantly enhance the efficiency of hot transitions in the weakly exergonic region.11

Figure 2. Molecular structures of the investigated compounds.

this trend was observed in [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ complex where a considerable decrease of the relaxed electron transfer product yield in more viscous solvent was discovered.22 On the other hand, investigations on [(tpy)RuII(tpy-phtpy)CoIII(tpy)]5þ complex have shown a weak opposite dependence of electron transfer product yield on solvent viscosity at room temperature. Nevertheless, in this compound there is a fall of the product yield at lower temperatures22 which can also be associated with the increase of solvent viscosity. These results and the revealed nonexponential kinetics of the ground state recovery (two time scales differ by a factor on the order of 10)21 are a real challenge to electron transfer theory. The aim of this paper is an elucidation of a detailed microscopic mechanism of intramolecular ultrafast forward and backward electron transfer in excited RuII-CoIII compounds. The questions to be answered are the following: (i) Can the theory quantitatively reproduce the excited state electron transfer and the following product decay kinetics? (ii) Why do two complexes with similar parameters demonstrate essentially different behaviors? (iii) Can the theory describe the influence of the solvent relaxation time on the product yield? (iv) What is the nature of the two time scales in the product decay? To answer these questions a stochastic four-level model involving the fast relaxation of intramolecular high frequency vibrations and the explicit description of medium relaxation is elaborated. Intramolecular high frequency modes are well-known to play an important role in highly exergonic electron transfer reactions in the Marcus

II. ELECTRON TRANSFER MODEL A scheme of the electronic states involved in photoinduced electron transfer processes in [(tpy)RuII(L-L)CoIII(tpy)]5þ (LL = tpy-ph-tpy and tpy-tpy) complexes is shown in Figure 3. Excitation of the metal-to-ligand charge transfer (MLCT) band by a short laser pulse leads to population of the 1MLCT(Ru) state which undergoes fast exothermic intersystem crossing to the triplet state 3MLCT(Ru) within approximately 100 fs. Subsequently, the excited state electron transfer (EET) from 3 MLCT(Ru) proceeds to the state [(tpy)2 Ru III (L-L)2 Co II (tpy)]5þ in 1-2 ps. This channel of deactivation of the excitation energy prevails over the competing channel of energy transfer. 22 Nonetheless, direct measurements performed in ref 22 have shown a rather low quantum yield of thermalized EET product Φ EET counting from 0.85 to 0.2 depending on solvent and temperature. To explain this experimental result, it was supposed that nonequilibrium products of EET undergo ultrafast backward electron transfer to the lower-lying triplet state [(tpy)1 RuII(L-L)3 CoIII (tpy)]5þ (see Figure 3) in parallel with nuclear relaxation. This process was interpreted in ref 22 as a hole transfer (HT) from 2 RuIII with dπ 5 electronic configuration to 2 CoII with dπ 6 dσ* to form 3 (dπ5 dσ*) of the Co III moiety. The thermal HT from the equilibrium [(tpy)2 RuIII (L-L)2 Co II (tpy)]5þ state is expected to be slow due to the large energetic barrier. On the other hand, the return electron transfer (RET) to the ground state of the complex involves reorientation of electronic spins and turns out to be even slower. 22 Hereafter we shall use the term “backward electron transfer” to mean both HT and RET. Electron transfer reactions in non-Debye solvents with complex nonexponential functions of relaxation,23,26 such as acetonitrile (ACN) and butyronitrile (BN), are generally described in terms of multidimensional free energy surfaces corresponding to different electronic and vibronic states of the chemical system. The relaxation function of solvent polarization X(t) is usually 1463

dx.doi.org/10.1021/jp108607t |J. Phys. Chem. A 2011, 115, 1462–1471


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represented as a sum of N exponentials:27 XðtÞ ¼

N X i¼1

xi e - t = τi

state of the complex (Ugr surface) as well as their vibrational sublevels in the form37,38 ð1Þ

In this case, each component in the sum can be associated with a separate solvent coordinate Qi with characteristic relaxation time τi. These solvent coordinates constitute N-dimensional configuration space of the problem. The quantities Qi are usually associated with different kinds of relaxation processes of the medium. Acetonitrile is generally described by two relaxation modes:26,28,29 the fast inertial mode with relaxation time scale τ1 = 0.19 ps and a slow diffusive mode with τ2 = 0.50 ps.10,30 The corresponding relaxation time scales in BN are τ1 = 0.19 ps and τ2 = 3.6 ps.22 In many polar solvents the multiexponential decay of autocorrelation function (eq 1) is achieved only at relatively long times and a considerable part of solvent relaxation is inertial.26,28,29,31 The autocorrelation function at this stage is better described by the Gaussian-type function, exp(-t2/τ2). The exponential and the Gaussian correlations reflect essentially different types of motion along the reaction coordinate.32 In particular, statistics of the reaction sink crossings are different in these cases, which in principle can change the reaction dynamics.33 In the case of strong electronic coupling these changes can be substantial, but in the opposite limit, when the Golden rule is applicable, the rate constant of thermal reaction does not depend on the dynamic properties of the solvent at all. Since this limit is operative for the systems considered, the kinetics of both EET and RET, proceeding in the thermal regime, are practically independent of the dynamic characteristics of the solvent. The behavior of the hot transition probability is quite different. This probability is weakly dependent on the reaction coordinate relaxation time in the strong electronic coupling limit (solventcontrolled regime) and is proportional to this time in the opposite limit.9 In the latter case, the hot transition probability is given by the Landau-Zener formula:34,35 W ¼

2πV 2 , pjQ_ j

Ue2 ¼ ðnÞ

Uet ¼

Q 12 Q 22 þ 4Er1 4Er2

ð5Þ

ðQ 1 - 2Er1 Þ2 ðQ 2 - 2Er2 Þ2 þ þ ΔGet þ npΩ ð6Þ 4Er1 4Er2 Ue1 ¼

Q 12 Q 22 þ þ ΔG21 þ mpΩ 4Er1 4Er2

ð7Þ

UgrðlÞ ¼

Q 12 Q 22 þ þ ΔG20 þ lpΩ 4Er1 4Er2

ð8Þ

ðmÞ

Here Ω is the frequency of the effective intramolecular vibrational mode, n, m, and l are the quantum numbers of vibrational sublevels, and Eri’s are the reorganization energies of solvent modes. The weight xi in eq 1 gives a contribution of the ith mode to the total reorganization energy of the solvent Erm: xi ¼ Eri = Erm ,

Erm ¼ Er1 þ Er2

ð9Þ

ΔG et is the free energy change associated with the excited state electron transfer Ue2 f Uet, the vertical free energy shifts between U e2 and Ue1 , and between U e2 and Ugr surfaces are denoted by ΔG 21 and ΔG20 , respectively. The temporal evolution of the chemical system is described by the set of differential equations36,39,40 X DFe2 ðnÞ ê2 Fe2 ¼ L kEET n ðFe2 - Fet Þ Dt n

ð10Þ

ðnÞ

DFet ðnÞ êt FetðnÞ þ kEET ¼ L n ðFe2 - Fet Þ Dt X ðnÞ ðmÞ kHT nm ðFet - Fe1 Þ m

W,1

ð2Þ

X l

where V is the electronic coupling, p is Planck’s constant/2π, and · |Q| is the velocity of the wave packet passing through the term crossing point. For the exponential correlation function9 jQ_ j ¼ jEr þ ΔGj = τ and for the Gaussian correlation function pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jQ_ j ¼ 2jEr þ ΔGj lnðjEr þ ΔGj = 2Er Þ = τ

ðn þ 1Þ

ðnÞ

Fet

Fet

τv

τv

ðn þ 1Þ

ð11Þ

ðnÞ

ðmÞ ðm þ 1Þ ðmÞ X DFe1 Fe1 Fe1 ðmÞ ðnÞ ê1 FðmÞ ¼ L kHT e1 nm ðFe1 - Fet Þ þ ðm þ 1Þ - ðmÞ Dt τv τv n

ð3Þ

ð12Þ DFðlÞ gr

ð4Þ

where Er, ΔG, and τ are the medium reorganization energy, the free energy change of the reaction, and the autocorrelation function decay time. For the systems considered here, the hot transitions proceed mainly in the region of small |ΔG| < Er, where the difference between eqs 3 and 4 is insignificant. This allows the whole relaxation process to be described in terms of several diffusive modes. Henceforward, we shall approximate the solvent relaxation function with eq 1. Adopting the four-level model for the chemical system under study, we introduce the diabatic free energy surfaces for the 3 MLCT(Ru) state (Ue2 surface), the EET product state (Uet surface), the triplet excited state (Ue1 surface), and the ground

ðnÞ

ðlÞ kRET nl ðFet - Fgr Þ þ

Dt

^gr FðlÞ ¼ L gr -

X n

ðnÞ

ðlÞ kRET nl ðFgr - Fet Þ þ

Fðlgr þ 1Þ

FðlÞ gr

τv

τv

ðl þ 1Þ

ðlÞ

ð13Þ

(n) (m) (m) Here Fe2 = Fe2(Q 1,Q 2,t), F(n) et = Fet (Q 1,Q 2,t), Fe1 = Fe1 (Q 1, (l) = F (Q ,Q ,t) are the probability distribution Q2,t), and F(l) gr gr 1 2 ê2, L êt, functions on Ue2, Uet, Ue1, and Ugr surfaces, respectively; L ^gr are the Smoluchowski diffusion operators: ê1, and L L " # 2 1 D D 2 ê2 ¼ L ê1 ¼ L ^gr ¼ L 1þQ1 þ ÆQ 1 æ τ1 DQ 1 DQ 21 " # 1 D D2 2 þ 1 þ Q2 þ ÆQ2 æ ð14Þ τ2 DQ 2 DQ 2 2

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Table 1. Experimentally Measured Rate Constants k1, kr, and kβ, Weights fr and fβ, and Relaxed EET Product Yield ΦEET complex [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ [(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ a

solvent

k1, 1010 s-1

ACNa

100

BNb

100

kR, 1010 s-1

fR

kβ, 1010 s-1

fβ

ΦEET

7.6

0.074

3.0

0.926

0.65

0.59

2.8

0.41

0.53

52

ACNa

30

5.6

0.167

0.45

0.833

0.39

BNb

32

8.1

0.45

0.49

0.55

0.41

Reference 20. b Reference 21.

Table 2. Electron Transfer Parameters (in electronvolts) for RuIICoIII Complexes in ACN22a complex II

III

[(tpy)Ru (tpy-tpy)Co (tpy)]

5þ

[(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ a

ΔGet

Erv

Erm

ΔGHT

ΔGRET

-1.0

0.48 ( 0.15

0.72 ( 0.15

-0.15

-0.97

-1.0

0.48 ( 0.15

0.85 ( 0.15

-0.15

-0.99

The free energies of HT and RET are ΔGHT = ΔG21 - ΔGet and ΔGRET = ΔG20 - ΔGet.

" # 2 1 D D 2 êt ¼ 1 þ ðQ1 - 2Er1 Þ þ ÆQ1 æ L τ1 DQ 1 DQ 1 2 " # 1 D D2 2 þ 1 þ ðQ2 - 2Er2 Þ þ ÆQ2 æ τ2 DQ 2 DQ 2 2

ð15Þ

with ÆQi2æ = 2ErikBT being the dispersion of the equilibrium distribution along the ith solvent coordinate. kB is the Boltzmann constant, and T is the temperature. The model eqs 10-13 imply a single-quantum mechanism of vibrational relaxation of the type |næ f |n - 1æ with the rate constant 1/τ(n) v . Electronic transitions between the excited state Ue2 and the nth vibrational sublevel of the EET product state U(n) et are described by the rates11 kEET ¼ n

2πVEET 2 Fn ðSEET Þn e - SEET ðnÞ δðUe2 - Uet Þ, , Fn ¼ p n! EEET SEET ¼ rv ð16Þ pΩ

The HT processes of the form rates

U(n) et

T

U(m) e1

proceed with the

2πVHT 2 Fnm ðnÞ ðmÞ δðUet - Ue1 Þ, p 2 3 ffi n þ m - 2r 2 minðn X, mÞ ð- 1Þn - r ðpffiffiffiffiffiffiffi S Þ HT 5, ¼ expf- SHT gn!m!4 r!ðn - rÞ!ðm - rÞ! r¼0 kHT nm ¼

Fnm

SHT ¼

EHT rv pΩ

ðmÞ

¼ FðlÞ gr ðQ 1 , Q 2 , t ¼ 0Þ ¼ 0

ð20Þ

The set of eqs 10-13 with the initial conditions, eqs 19 and 20, has been solved numerically employing the Brownian simulation algorithms developed earlier.11,41-43 The population kinetics of the excited state, Pe2(t), and the EET product state, Pet(t), were calculated as follows: Z Pe2 ðtÞ ¼ Fe2 ðQ 1 , Q 2 , tÞ dQ 1 dQ 2 , Pet ðtÞ Z

ð17Þ

Fet ðQ 1 , Q 2 , tÞ dQ 1 dQ 2

ð21Þ

In section III these quantities are fitted to the experimental data.

2πVRET 2 Fnl ðnÞ δðUet - UgrðlÞ Þ, kRET ¼ nl p 2 3 ffi n þ l - 2r 2 minðn X, lÞ ð - 1Þn - r ðpffiffiffiffiffiffiffiffi S Þ RET 5, Fnl ¼ expf - SRET gn!l!4 r!ðn - rÞ!ðl - rÞ! r¼0 ERET rv pΩ

ðnÞ

Fet ðQ 1 , Q 2 , t ¼ 0Þ ¼ Fe1 ðQ 1 , Q 2 , t ¼ 0Þ

¼

(l) For RET from U(n) et to Ugr one obtains

SRET ¼

where Fnm are the Franck-Condon factors; Sk and Ekrv (k = EET, HT, and RET) are the Huang-Rhys factors and the reorganization energies of the intramolecular high frequency mode for the corresponding electronic transitions. Now we are going to formulate the initial conditions for eqs 10-13. At the first stage of the reaction (1MLCT(Ru) and 3 MLCT(Ru) formation), the electronic distribution is shifted from RuII to ligand. Since this shift is about 15% of the metal-tometal distance,20 one may suppose the solvent polarization to be disturbed by MLCT only slightly. This allows us to assume the initial distribution on the second excited term Ue2 to be equilibrium: Fe2 ðQ 1 , Q 2 , t ¼ 0Þ " # 1 Q 12 Q 22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ¼ ð19Þ 2ÆQ 1 2 æ 2ÆQ 2 2 æ 2π ÆQ 1 2 æÆQ 2 2 æ

ð18Þ

III. RESULTS OF SIMULATIONS AND DISCUSSION OF THE REACTION MECHANISMS III.1. Experimental Data To Be Fitted. The aim of numerical simulations is to reproduce the EET, HT, and RET kinetics measured in ACN and BN solutions at room temperature (kBT = 0.025 eV). Experiments with [(tpy)RuII (L-L) CoIII(tpy)]5þ complexes revealed a quasi-exponential decay of the excited state absorption for MLCT state with the rate k1 and a biexponential recovery of the MLCT absorption band with the rates kR and kβ 1465



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and the weights fR and fβ, respectively. The results of the experiments20-22 are summarized in Table 1. These data allow restoring the kinetics of populations for the initial excited state, Pe2(t), and the EET product state, Pet(t), as follows: Pe2 ðtÞ ¼ e - k1 t Pet ðtÞ ¼ A½fR ðe - kR t - e - k1 t Þ þ fβ ðe - kβ t - e - k1 t Þ

ð22Þ ð23Þ

Parameter A is determined from the condition max{Pet(t)} = ΦEET. Here the intersystem crossing between Ue1 and Ugr terms is supposed to proceed faster than HT. The solution of eqs 10-13 with the initial conditions 19 and 20 was fitted to the restored population kinetics, eqs 22 and 23. In simulations we used dielectric relaxation parameters of ACN reported in ref 10, x1 = 0.5, x2 = 0.5, τ1 = 0.19 ps, and τ2 = 0.5 ps, and those of BN reported in ref 22, x1 = 0.75, x2 = 0.25, τ1 = 0.19

ps, and τ2 = 3.6 ps. The frequency and the relaxation time of the effective quantum intramolecular vibrational mode are taken pΩ = 0.1 eV and τv = 30 fs. There are also reliable estimations for some energetic parameters of the forward and backward electron transfer given in ref 22. These data are collected in Table 2. In simulations part of the parameters (Erm, ΔGet, ΔG21) are varied within the limits of estimation errors; the others (VEET, VHT, VRET, etc.) are taken as free fitting parameters. Fitting the forward and backward electron transfer kinetics to the experimental data is carried out for both [(tpy)RuII(LL)CoIII(tpy)]5þ complexes in ACN and BN solvents. This is done in two steps: first we fitted the kinetics of forward electron transfer, EET, and then we fitted the kinetics of recombination through the HT and RET channels as shown in Figure 4. III.2. Fitting to the [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ Compound Kinetics. Fitting results for the [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ compound are displayed in Figure 5. The two upper panels represent the population kinetics of the excited state, Pe2(t), while the two bottom panels show the population kinetics of the EET products, Pet(t). The left and the right panels correspond to the different solvents—ACN and BN, respectively. The best fit parameters are listed in Table 3. Table 3. Best Fit Parameters bridging ligand (L-L) parameter

Figure 4. Cuts in the free energy surfaces of the diabatic electronic states and their vibrational repetitions for [(tpy)RuII(L-L)CoIII(tpy)]5þ (L-L = tpy-tpy, tpy-ph-tpy) complexes. EET and RET primarily proceed in thermal regime, while HT mainly occurs at the hot stage.

tpy-tpy

tpy-ph-tpy

Erm

0.65

0.90

ΔGet

-0.84

-0.99

ΔG21

-1.15

-1.10

EEET rv

0.48

0.20

EHT rv

0.48

0.48

ERET rv

0.48

0.20

VEET VHT

0.0115 0.0092

0.0047 0.0203

VRET

0.0009

0.0006

Figure 5. Population kinetics of the second excited state, Pe2(t), and the EET product state, Pet(t), for [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ compound in ACN and BN solvents. Solid lines are the numerical data; circles represent the reconstructed experimental kinetics. The best fit parameters are given in Table 3. 1466


The Journal of Physical Chemistry A One can easily see from Figure 5 that the results of simulations (solid curves) give a rather good quantitative description of the experimental data (open circles) at the whole time interval. First of all, simulations in ACN and BN were able to reproduce the quasi-exponential kinetics of forward electron transfer (compare upper panels in Figure 5) with the rate weakly dependent on solvent viscosity. This is a clear indication that EET proceeds in nonadiabatic regime and the nuclear subsystem remains quasi-equilibrium in the course of the reaction. On the other hand, population kinetics of the EET products show significant changes upon moving from the fast solvent (ACN, Figure 5B) to the slower one (BN, Figure 5D). Both experimental data and simulation results demonstrate (1) a considerable decrease in quantum yield of EET products (∼20%) and (2) more pronounced deviations of Pet(t) from a single-exponential decay in BN compared with ACN. At first thought these phenomena indicate that decay of the primary EET products proceeds in the so-called solvent-controlled regime, where the rate of electron transfer is known to depend on the solvent relaxation time scale.39 This hypothesis is not however confirmed by numerical simulations since the coupling parameters VHT and VRET are found to be even smaller than VEET (see Table 3). Simple estimations show that backward electronic transitions in this system proceed nonadiabatically in the same way as the forward ones. Since dielectric characteristics of the solvents are different (εs is equal to 36 and 24 and εop is equal to 1.78 and 1.91 for ACN and BN, correspondingly), one should expect a 10% decrease of Erm in BN compared to that in ACN according to the Marcus equation:44 ! e2 1 1 1 1 2 Erm ¼ þ ð24Þ RA RD RAD 2 εs εop where RA and RD are the effective radii of the acceptor and donor, RAD is the charge transfer distance, and εs and εop are the static and optical dielectric constants of solvent. However, we failed to fit the reaction kinetics in both solvents satisfactorily keeping such a relation between the solvent reorganization energies. A good fit was achieved with equal values of Erm in ACN and BN only. The reasons for this result are not quite clear. We suggest that an explanation of this phenomenon should involve a detailed examination of the molecular structure of the solvent that is totally ignored in eq 24. III.3. Reaction Mechanisms in the [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ Compound. To explain the features of the experimentally measured reaction kinetics, one should take into account that, in contrast to EET, HT in this chemical system proceeds at least partly in the nonequilibrium regime. The initial wave packet formed by EET gradually depletes in the course of its relaxation down to the bottom of the EET product well, U(n) et , due to backward transitions (mainly HT; see Figure 4). This type of transition is known as “hot” electron transfer. Since hot transitions are limited by the time of solvent relaxation, their efficiency is characterized by the transition probability but not the rate. This probability directly determines the yield of thermalized EET product. Unlike the thermal rate constant, the transition probability rises with the solvent relaxation time in the nonadiabatic regime and becomes nearly independent of it in the solvent-controlled regime.9,12 This means that at the hot stage the kinetics of the nonadiabatic chemical reaction are determined not only by term geometry and by electronic coupling between them. The key role in this process belongs to relaxation characteristics of the solvent as well (see Figure 4).

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Figure 6. Trajectories of nonequilibrium wave packets on the EET product free energy surface U(n) et . The trajectories (eqs 27and 28) start at (n) the (Q(n) 1 , Q2 ) points and go down to the term minimum (2Er1, 2Er2). Uet and Ue1 free energy surfaces are shown by colored and black dashed lines, respectively. Thin straight lines are the positions of the HT sinks (eq 29). Blue and red colors correspond to ACN and BN solvents.

According to the experimental data from refs 10 and 22, the time scales of the so-called diffusive component of dielectric relaxation (the τ2 parameter) in ACN and BN differ by a factor of an order of magnitude. This makes a great impact on the time of residing of the nonequilibrium wave packet in the area of the most intensive electronic transitions. As a result the yield of thermalized EET products in BN is sufficiently lower than that in ACN (see Figure 5), contrary to the rate of initial recombination at the time interval up to ∼5-8 ps, which is faster in BN than in ACN. The differences of the Pet(t) kinetics in ACN and BN solvents can be better appreciated if we consider the motion of nonequilibrium wave packets on the Uet(Q1,Q2) free energy surface. An illustration is given in Figure 6, where the blue and red colors correspond to ACN and BN, respectively. The wave packets produced by the transitions with the creation of n vibrational quanta appear on the EET product surface in the neighborhood of the saddle points of lower branches of the free energy surfaces Ue2 and U(n) et : ðnÞ

¼

Er1 ðErm þ ΔGet þ npΩÞ Erm

ð25Þ

ðnÞ

¼

Er2 ðErm þ ΔGet þ npΩÞ Erm

ð26Þ

Q1 Q2

These saddle points are shown by small colored circles in Figure 6. 1467



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One can obtain from the equations of motion eqs 10-13 that the gravity centers of the nonequilibrium wave packets in the EET product state move to the U(n) et surface minimum along the parametric curves of the form ðnÞ ~ ðnÞ Q 1 ðtÞ ¼ 2Er1 þ ðQ 1 - 2Er1 Þ expð- t = τ1 Þ

ð27Þ

ðnÞ ~ ðnÞ Q 2 ðtÞ ¼ 2Er2 þ ðQ 2 - 2Er2 Þ expð- t = τ2 Þ

ð28Þ

In Figure 6 these trajectories are shown by thick colored lines that (n) start at the (Q(n) 1 , Q2 ) point and go down to the term minimum (2Er1, 2Er2). The hot HT occurs when the wave packets cross the HT reaction sinks located along the intersections between U(n) et and U(m) e1 free energy surfaces. These intersections are given by the equation Q 1 þ Q 2 ¼ Erm þ ΔGet - ΔG21 þ ðn - mÞpΩ

ð29Þ

and are shown in Figure 6 as thin straight lines corresponding to n = 0 and several m values indicated in the figure. The dynamics of the EET product population includes three stages. At the first stage Pet(t) rises. This stage lasts while 3MLCT state decay occurs. At the second stage nonequilibrium HT predominantly occurs, and at the last stage RET in the thermal regime proceeds. In ACN the time scale of solvent relaxation is shorter than that of EET so that the second stage is hardly seen in population kinetics since it terminates almost simultaneously with the first one. In BN the inequality τ2 . 1/k1 is fulfilled that leads to a prominent nonequilibrium stage (see Figure 5). Moreover, in BN, in contrast to ACN, the particles accumulate in the vicinity of the trajectory sharp bend (see Figure 6), since the motion along the Q2 coordinate is much slower than both 3 MLCT state decay and Q1 mode relaxation. This results in two important consequences. First, the hot HT proceeds more effectively during EET that reduces the yield ΦEET. Second, the high concentration of particles in the vicinity of the strong HT sinks accounts for the considerably higher rate of Pet(t) decay at the nonequilibrium stage in BN. This can be regarded as the main reason for conspicuous biexponential Pet(t) decay kinetics observed in the [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ compound in BN. The two stages of the decay are also present in ACN, but the weight of the fast stage is rather small and it is hardly visible in Figure 5B. It should be noticed that, generally speaking, the quantity ΦEET cannot be interpreted as the quantum yield of the thermalized EET products since the initial stage of EET product population decrease proceeds in the nonthermal regime. In ACN solution ΦEET is close to the quantum yield of the thermalized EET products, but in BN there is a large difference between them. III.4. Quantum Yields of Vibronic Products in Forward and Backward Electron Transfer. It is also instructive to consider the distribution of quantum yields of forward and backward electron transfer through the EET, HT, and RET channels. The quantum yield of EET products in nth vibrational state is calculated as follows: Z Z ðnÞ YEET ¼ dt dQ1 dQ2 kEET n ðQ 1 , Q 2 Þ½Fe2 ðQ1 , Q2 , tÞ ðnÞ

- Fet ðQ 1 , Q 2 , tÞ

ð30Þ

Figure 7. Quantum yields of vibronic products produced through the EET channel (A) and the HT and RET channels (B) in [(tpy)RuII(tpytpy)CoIII(tpy)]5þ compound. The values for quantum numbers n, m, and l are indicated on the plot. The results in ACN are shown as white bars; those in BN are shown as gray bars.

while the corresponding quantum yields of HT and RET are Z Z X ðmÞ ðnÞ kHT YHT ¼ dt dQ 1 dQ 2 nm ðQ1 , Q2 Þ½Fet ðQ 1 , Q 2 , tÞ n ðmÞ - Fe1 ðQ1 , Q 2 , tÞ ðlÞ

YRET ¼

Z

Z dt

dQ 1 dQ 2

X

ð31Þ ðnÞ

kRET nl ðQ 1 , Q 2 Þ½Fet ðQ 1 , Q 2 , tÞ

n

- FðlÞ gr ðQ1 , Q2 , tÞ

ð32Þ

These quantities indicate the roles of different reaction channels as well as intramolecular high frequency vibrations in forward and backward electron transfer. (n) Histograms in Figure 7 show the distribution of YEET (m) (l) (Figure 7A), YHT and YRET (Figure 7B) in ACN (white bars) and BN (gray bars). The abscissa in these plots is the collective solvent coordinate Q = Q 1 þ Q 2 defining the positions of the reactive sinks. One can easily see from these results that Y(n) EET in this chemical system is almost independent of solvent. This actually means that EET keeps a quasi-equilibrium distribution of particles on the Ue2 electronic surface both in ACN and in BN. On the other hand, there is a considerable difference between the (l) quantities Y(m) HT and YRET in these solvents (see Figure 7B). In particular, the efficiency of the RET channel is more prominent in acetonitrile, while the HT channel becomes more productive in BN. These results reflect the differences in solvent relaxation functions and can be regarded as a manifestation of nonequilibrium effects in backward electron transfer. 1468



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Figure 8. The same as in Figure 5, but for [(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ compound. The best fit parameters are given in Table 3.

III.5. Fitting to the [(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ Compound Kinetics. The results of the fitting for [(tpy)RuII(tpy-

ph-tpy)CoIII(tpy)]5þ compound are displayed in Figure 8. The arrangement of panels here is the same as in Figure 5. The best fit parameters are given in Table 3. As shown in Figure 8, the reaction kinetics is well reproduced in ACN and the quantum yields of EET products are well reproduced in both solvents. The kinetics of Pet(t) decay in BN (Figure 8D) however can not be reproduced by the present model. The principal distinction between the two compounds considered is the dependence of ΦEET on the solvent viscosity. For the second compound, in contrast to the first one, ΦEET weakly rises with viscosity increase.22 This is possible if the probability of hot HT does not increase with viscosity. Simulations can reproduce such a behavior only if the HT terminates at the stage of the fast solvent mode relaxation. This requires considerably larger values of the medium reorganization energy to provide leaving of the wave packet from the HT sink region during relaxation of the fast solvent mode. As a result, the fitting gives the value Erm = 0.90 eV instead of Erm = 0.65 eV in the previous case. This difference appears rather natural because in [(tpy)RuII(tpy-phtpy)CoIII(tpy)]5þ the distance of electron transfer is considerably larger (1.3 versus 0.94 nm).22 On the other hand, the suppression of HT at the stage of slow solvent mode relaxation implies suppression of the nonequilibrium stage of the EET product depopulation that leads to nearly exponential kinetics while the experimental data demonstrate pronounced biexponential decay kinetics. The discrepancy between experimental and simulation kinetics at the initial time interval up to t ∼ 20 ps cannot be eliminated by involving a nonequilibrium stage since the slow mode in BN completes its relaxation earlier at t ∼ 8 ps as in the previous compound. If we assume that the decay with the rate constant ka = 8.1 1010 s-1 is associated with the nonequilibruim regime, then a relaxation mode with a time constant of τ3 ≈ 10 ps should be involved in the electron transfer. This mode may be connected with some kind of structural changes in the system10 (complex þ solvent) and cannot be attributed to the medium relaxation itself.

Figure 9. The same as in Figure 7, but for [(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ compound.

An additional support to the above picture may be obtained from histograms in Figure 9 that show the distributions of vibronic products of EET, HT, and RET for the [(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ compound. A small variation of Y (n) EET with solvent is caused by alteration of the medium reorganization energy. On the other hand, the distributions of backward electron transfer products over HT and RET channels are practically independent of solvent. This feature may be elucidated in terms of the nonequilibrium wave packet propagation. As shown in 1469



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alteration of ΔGHT raise the activation barrier between U(0) et and U(0) e1 free energy surfaces from Ea = 0.044 eV = 1.8kBT to Ea = 0.173 eV = 6.9kBT. This is sufficient to suppress the thermal as well as nonthermal HT at the stage of diffusive mode relaxation. Distinctions in reaction kinetics for the two complexes also manifest themselves in different values of electron transfer parameters indicated in Table 3. In particular, one may expect smaller values of all matrix coupling elements for the longer complex due to the larger distance between the metallic centers. This is true for VEET and VRET, but not for VHT as shown in Table 3. The larger value of the best fit parameter VHT in [(tpy)RuII(tpy-phtpy)CoIII(tpy)]5þ complex stems from the smaller value of ΦEET. Indeed, in the framework of the present model, the only way to reduce maximum concentration of EET products is to raise the magnitude of V HT . Although such a result cannot be expected if the reaction is considered to be through space purely, in principle it is possible in the case of through ligand interaction. The large value of V HT , obtained from the fitting, may also indicate a superexchange interaction via the p y orbitals of the bridging ligands of tpy-tpy and tpy-ph-tpy to be operative in these compounds. 22

Figure 10. Accumulation kinetics of backward electron transfer products through HT and RET channels. Panels A and B correspond to [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ and [(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ compounds, respectively.

Figure 6B, the wave packet passes the HT sinks during fast solvent mode relaxation. Since τ1 is identical in both solvents, the efficiency of the HT channel appears to be the same. Therefore, there is a great difference in the pictures of electron transfer in the [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ and [(tpy)RuII(tpy-ph-tpy)CoIII(tpy)]5þ compounds. First, the efficiency of HT is considerably higher in the former compound while the efficiency of RET is larger in the latter. Second, the vibronic distribution of HT products is almost solvent independent in the latter compound in contrast to the former. Third, in [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ the efficiency of HT sinks monotonically increases with the number m decrease while in second compound it has a pronounced maximum reflecting the distribution of FranckCondon factors for 0 f m transitions as well as depletion of the nonequilibrium wave packet in the course of its relaxation down to the U(n) et term minimum. The kinetics of HT and RET product formation are also significantly different for the two compounds (see Figure 10). Comparing panels A and B of Figure 10, one can easily see that the first compound appears to be much more sensitive to the dynamic properties of the solvent. Moreover, in [(tpy)RuII(tpy-phtpy)CoIII(tpy)]5þ the HT proceeds entirely at the nonequilibrium stage (Figure 10B) whereas in [(tpy)RuII(tpy-tpy)CoIII(tpy)]5þ a considerable part of HT occurs in thermal regime (Figure 10A). In BN the thermal part amounts to =25% and reaches =50% in ACN. Different behavior of the compounds is mainly attributed to the differences in solvent reorganization energy, Erm. The increase of Erm from 0.65 to 0.90 eV and simultaneous

IV. CONCLUDING REMARKS The stochastic four-level model elaborated in this paper gives evidence in favor of the guess on a double-channel character of backward ET made in ref 22. Both the HT and RET channels appear to play an important role in these processes, but operate differently: the HT proceeds predominantly in the nonequilibrium regime, while the RET occurs largely under equilibrium conditions. In general, the efficiency of hot electronic transitions in the nonadiabatic regime depends on the dynamic characteristics of the solvent. As a result, one should expect a variation of EET product yield with solvent viscosity. This expected variation was distinctly observed in the [(tpy)1RuII(tpy-tpy)3CoIII(tpy)]5þ compound, but in the second compound a weak opposite trend was detected. This unexpected dependence is elucidated in terms of two time scales of the solvent relaxation. The HT efficiency in the [(tpy)1RuII(tpy-ph-tpy)3CoIII(tpy)]5þ compound is practically solvent independent since HT terminates at the stage of relaxation of the fast solvent mode and the time scale of this relaxation is independent of the solvent viscosity. A small variation of energetic electron transfer parameters due to alteration of the solvent dielectric permittivity is responsible for a minor increase of ΦEET in BN. In ref 22 the temperature dependence of the EET product yield and the RET kinetics were studied for the [(tpy)1RuII(tpyph-tpy)3CoIII(tpy)]5þ compound in BN. The fall of the yield from ΦEET = 0.41 at 297 K to ΦEET = 0.21 at 180 K was observed. This result is in qualitative agreement with the current theory since the solvent relaxation decelerates at lower temperatures and the efficiency of hot HT increases. However, to examine the possibility of quantitative description of these results, the current theory needs data on the temperature dependence of the solvent relaxation time scales. Unfortunately, such information is absent in the literature. The model provides a quantitative description of forward and backward electron transfer kinetics in ruthenium(II)-cobalt(III) compounds. In [(tpy)1RuII(tpy-tpy)3CoIII(tpy)]5þ the model allows reproducing the experimentally observed quasiexponential kinetics of the excited state decay and the biexponential kinetics of the ground state recovery in both ACN and 1470


The Journal of Physical Chemistry A BN including the variation of the electron transfer product yield with solvent. In [(tpy)1RuII(tpy-ph-tpy)3CoIII(tpy)]5þ the exponential kinetics of EET and the nonexponential kinetics of backward electron transfer are well reproduced in ACN. The model is capable of reproducing a small rise of the electron transfer product yield in passing from ACN to more viscous solvent, BN, but fails to reproduce the short time kinetics of backward electron transfer in BN. This discrepancy may reflect some kind of relaxation process that occurs in the complex being dissolved in BN but does not occur in ACN. Unsatisfactory fitting results for the electron transfer kinetics in the long bridge complex are enhanced by apparent inconsistences in the best fit parameters. First, the larger value of HT electronic coupling VHT in this complex compared to that in the short complex (Table 3) has been obtained, although one should expect the opposite result, due to the larger distance between the metallic centers. Second, the vibrational reorganization energy for HT, ErvHT, in [(tpy)1RuII(tpy-ph-tpy)3CoIII(tpy)]5þ is expected to be much smaller than those for EET and RET, EEET rv and ERET rv , because the latter processes involve the change in electron occupation in the antibonding dσ orbitals whereas the former ET process occurs between nonbonding dπ orbitals in Ru(II) and Co(III).22 However, the trend of the obtained parameters considerably disagrees with these expectations, and cannot be eliminated within the framework of the present model. This trend is a direct consequence of a smaller value of ΦEET in the longer compound that implies a larger probability of hot HT. These drawbacks call for a further improvement of the model. Such an improvement may involve reversible doublet-quartet intersystem crossing21 (ISC) 2CoII T 4CoII omitted in this model. A good fit of the model to the kinetics observed in the [(tpy)1RuII(tpy-tpy)3CoIII(tpy)]5þ compound implies a minor role of ISC in it. On the other hand, in the [(tpy)1RuII(tpy-phtpy)3CoIII(tpy)]5þ compound the RET kinetics is considerably slower, so ISC may compete with RET. Consideration of ISC in the longer compound allows fitting biexponential decay of the EET products. This however does not change the relation between the parameters VHT in the two compounds as well as EET RET in the longer the relation between EHT rv and both Erv and Erv compound. Expected relations between these parameters can be obtained if one suggests an existence of another ground state recovery channel competing with the mechanism considered. The excitation energy transfer (EnT) can play a role in such a channel.22 If this channel is more productive in the [(tpy)1RuII(tpy-phtpy)3CoIII(tpy)]5þ compound, then the probability of hot HT can be lower than that in [(tpy)1RuII(tpy-tpy)3CoIII(tpy)]5þ. This could result in the expected relation between the parameters.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the Ministry of Education and Science of the Russian Federation (Contract No. P1145) and the Russian Foundation for Basic Research (Grant 10-03-97007). ’ REFERENCES

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Kinetics of Nonequilibrium Electron Transfer in Photoexcited

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