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Intramolecular electron-transfer dynamics in the inverted regime: quantum mechanical multi-mode model including dissipation
CHEMICAL
30 August 1996
PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 259 (1996) I 13-118
Intramolecular electron-transfer dynamics in the inverted regime: quantum mechanical multi-mode model including dissipation Brigitte Wolfseder, Wolfgang Domcke Institute of Physical and Theoretical Chemistry, Technical Unioersity of Munich, D-85747 Garching, Germany Received 19 April 1996; in final form 20 June 1996
Abstract
A model of ultrafast electron transfer in condensed media is proposed; it includes electronic inter-state coupling, coupling of the electronic dynamics with several intramolecular modes, and damping of the vibrational motion due to interaction with a thermal environment. Exact numerical solutions of the equation of motion for the reduced density matrix are obtained for this dissipative multi-level system, employing the Monte Carlo wavefunction propagation method. The comparison of the electronic population dynamics of the multi-mode model with an effective single-mode description reveals the importance of multi-mode effects for ultrafast ET dynamics.
1. Introduction Electron-transfer (ET) reactions in the condensed phase have been of interest for decades, see, e.g., Refs. [1-3] for reviews. For many years, attention has been focused on the effect of solvent dynamics on ET reactions. More recently, an increasing number of experiments with femtosecond time resolution are providing evidence for the importance of intramolecular vibrational motion in ET processes, see, e.g., Refs. [4-9]. ET processes in the so-called inverted regime [1], in particular, seem to be controlled more by intramolecular vibrational dynamics rather than solvent effects [4-8]. Moreover, quantum beats reflecting damped coherent vibrational motion have been observed in several ET systems [5,10]. These observations point to the necessity of a fully quan-
tum mechanical and molecule-specific description of ET processes in the femtosecond time regime. Theoretical ET models which include intramolecular vibrational motion have been proposed by Sumi and Marcus [11] and Jortner and Bixon [12]. These early descriptions, which have been further elaborated recently [13-16], are based on the Fermi golden rule expression for the ET rate, and thus assume that vibrational energy and phase relaxation processes are fast compared to the ET reaction. In the case of ET dynamics in the inverted regime, where the reaction can be very fast and the vibrational excess energy is large [5-7], the golden rule description becomes questionable and should be replaced by a nonperturbative treatment. In recent years, the interplay of ET, coherent vibrational motion and dissipation by the solvent has
000%2614/96/$12.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PH S 0 0 0 9 - 2 6 14(96)0071 5-4
114
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113-118
been investigated via the direct numerical solution of the equation of motion for the reduced density matrix (RDM) for models which take into account intramolecular motion as well as coupling to a thermal environment [17-21]. Given the time-dependent RDM, the expectation values of observables pertaining to the system, e.g. electronic state populations or amplitudes of vibrational motion, can straightforwardly be evaluated. This fully quantum mechanical and nonperturbative approach appears to be particularly suitable for the treatment of ET in the inverted regime, where the electronic coupling may be strong and large amount of excess energy has to be dumped into vibrational motion, Since the computational cost of constructing the Redfield relaxation tensor [22] and propagating the RDM scales at least as N 2, where N is the dimension of the relevant Hilbert space of the system, the RDM approach has so far been limited to ET models with a single effective reaction mode [17-21]. It is to
pulses of finite duration, will be considered in a future publication.
be expected, however, that in many cases the ET in donor-acceptor systems involves a change of the equilibrium geometry of several intramolecular modes. A realistic description of the ET reaction may thus require the inclusion of several strongly coupled vibrational modes in the system Hamiltonian [23-25]. In the present work we demonstrate the computational feasibility of a multi-mode description of ET including dissipation by a thermal environment. We employ the so-called Monte Carlo wavefunction propagation method introduced by Dalibard et al., Zoller et al. and others for quantum optical master equations [26,27]. The wavefunction is propagated with a non-Hermitian effective Hamiltonian in the full direct-product Hilbert space of the system, including stochastic quantum jumps [26,27]. Expectation values of system operators are obtained by averaging over a sufficiently large number of realizations of the stochastic process [26,27]. In the present communication we assume, for simplicity and clarity, instantaneous excitation of the charge-transfer (CT)state by an ultrashort laser pulse, The observable of interest is the time-dependent population Pl(t) of the CT state or, equivalently, the ET rate -l~(t)/P~(t). The simulation of real-time pump-probe spectra, taking account of the preparation as well as the probing of the system by laser
Here Et denotes the vertical electronic excitation energy and g is the electronic inter-state coupling. The H i are vibrational Hamiltonians of the respectire electronic states. In terms of creation and annihilation operators pertaining to dimensionless normal modes of the electronic ground state, the H i read (h = 1)
2. Definition of the problem
2.1. Model Hamiltonian We consider a model of a typical photoinduced ET process, where a CT state I q~> is populated by optical excitation from the electronic ground state [ ~P0)- Internal conversion of the excited state to the ground state leads to ultrafast back-transfer of the electron. In adiabatic electronic representation, the system Hamiltonian is written as
Hs ~- E
I ~i)( Hi 4- ~i.l El)( ~°i l
i~ 0,1
+{I ~00)g(,pj 14,h.c.}.
(2.1)
M
Hi = E {wk(btkbk 4" ½) 4" ~i l/£k(bk 4-b~)/~/2"}. k=l
(2.2)
M is the number of vibrational modes to be included in the system Hamiltonian, tok is the vibrational frequency of the kth mode and Ks is the coupling parameter for the intra-state electronic-vibrational coupling, defined as the gradient of the excited-state potential-energy (PE) surface with respect to the kth mode. The dimensionless parameter Kk/to k determines the shift of the equilibrium geometry of the excited state in the kth mode. We shall assume that only few (say, three or four) vibrational modes are strongly coupled to the electronic transition. The remaining weakly or only indirectly coupled modes of the chromophore as well as the solvent degrees of freedom are considered as a thermal bath of harmonic oscillators. Assuming bilinear system-bath coupling and neglecting the ef-
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113 - 118
fect of g on dissipation, the bath variables can be traced out in the standard way, invoking perturbation theory for the system-bath coupling as well as the Markovian approximation [22,28]. The resulting equation of motion for the reduced density operator p(t) of the system reads 0 ~ p ( t ) = - i [ H s, p ( t ) ] + L ( p ) . (2.3) The relaxation operator L(p), expressed in terms of ground-state vibrational creation and annihilation operators, is given by
L ( p ) =½F ~., [~i>A(ij)(q~j],
(2.4)
115
Eq. (2.3) with the system Hamiitonian (2.1) and the relaxation operator (2.4) describes coupled electronic-state dynamics with M strongly coupled systern modes, which are in turn damped by the interaction with a thermal environment. The approximations which have been made in deriving this simple model are justified in the case of weak system-bath coupling and weak to moderate electronic inter-state coupling g. The range of validity of the model defined by Eqs. (2.3)-(2.5) extends, however, well beyond the golden rule regime. In the present application, we include three modes in the system Hamiltonian (M = 3). The vibrational
i,j= 0,1 M
((nk) + 1)(2bk p(O)b~ - b k*b, p(ij)
A(iJ) = E
~ a.o
k=l
\
\
/
~ 1.5 z30
-P(iJ)b~bk)
~ t.o
M
+ E (n,>(2b~
p(iJ)b, - bkb ~ p(ij)
~ o,s
k=l
- p(iJ)b k b~)
o.o
-4.o
M
"4- a i I E ' k=l
Kk
M
Kk
~
2.o
(b -bl)o (").
2.0
(2.5)
k=l
t.o
0.5
I ~:>
0.0
and
(n k) = [exp( WJkBT ) - 1] - '
•
g
Here p(ij) = < ~i I p
o.o
"~(iJ)(bCk -- bk)
~'~'O')k t~
E
'
-2.o
normal coordinate O~
-4.0
-2.0 0.0 normal coordinate Qz
2,0
(2.6)
is the thermal occupation distribution of the kth mode. In the derivation of Eqs. (2.3)-(2.5) the rotating-wave approximation for the system-bath coupiing has been invoked, which is justified for weak system-bath coupling [28]. In the case of strong system-bath coupling, non-secular relaxation terms have to be included. The corresponding relaxation operator, which is equivalent to the classical Brownian oscillator model [29] is also well known [30] and allows the modeling of overdamped vibrational motion. For simplicity, the damping rate F is assumed to be the same for all system modes,
>~ a.o g ~ 2.0 g t,0 o.o
-10.0 -8,0 -6.0 -4.0 -2.0
0.0
normal coordinate Qs
2,0
Fig. 1. Cuts through the S O and S~ diabatic PE surfaces of the three-dimensional ET model along each of the three normal coordinates. For each single mode, the crossing of the diabatic
potentials lies above the verticalexcitationenergy.
116
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113-118
frequencies are taken as w I = 0.26 eV, w 2 = 0.18 eV and to 3 = 0.07 eV. These values are representative for real molecules, as the entire range from low-frequency to high-frequency modes is covered. The electron-vibrational coupling constants are taken as K l 0.34 eV, ~c2 = 0.22 eV and K3 0.09 eV. The dimensionless geometry shift KJ % is thus approximately the same for all three system modes. The vertical energy gap is chosen as E l = 0.91 eV. The parameters have been chosen such that the model describes an ultrafast electron back-transfer process. The inter-state coupling constant is g = 0.06 =
ao ->~ ~.s g t.o
=
eV, and the damping rate of the oscillators is taken as F = 0.005 eV. With this choice of parameters, the approximations made in the derivation of the relaxation operator (2.4), (2.5) are well justified. Since temperature effects are not important for the strongly exothermic ET process considered here, the bath temperature T is chosen to be zero. Fig. 1 shows cuts through the diabatic PE surfaces of the S O and S, states along each of the three normal coordinates. It can be seen that for each mode the crossing of the diabatic surfaces lies above the vertical excitation energy E l . The minimum of the three-dimensional surface of intersection, however, lies very close to the minimum of the CT surface. We thus expect ultrafast surface-crossing dynamics in this three-mode model, In most theoretical models of ET which include intramolecular vibrational dynamics, it has been assumed that the intramolecular vibrational motion can be approximated by a single reaction mode, see, e.g., Refs. [11,12,17-21]. We are able to test the adequacy of this assumption by approximating the three-mode coupled-surface dynamics by an effective single-mode model. Following Refs. [31,32] the coupling constant and frequency of the effective mode are defined as
-~ 0.s O.
0.0 -4,o
-2.0
o.o
normal coordinate Q .
2,0
Fig. 2. Diabatic S O and Sj PE functions of the effective singlemode model. The surface crossing occurs close to the minimum of the S 1 PE function.
minimum of the latter. According to common reasoning, rapid ET is expected in this situation [14,16].
2.2. Numerical methods For tonian sented tronic
the numerical solution of Eq. (2.3) the Hamiland the reduced density operator are reprein the direct-product basis of diabatic elecstates [ ~o0), [ ~Pl) and harmonic-oscillator
states
I x.,)
for each mode
p(t) = ~.~ I ot)p~,a,(a'
l,
(2.9)
",'~' where I a) = I ~i)1X,,)
X,2)I Xo3).
(2.10)
Eq. (2.3) is solved with the initial condition
p(0) = I ~'l ) 10) 10) 10)(0 I(0 I(0 1( ~ I,
(2.11)
which corresponds to vertical excitation from the ground state to the CT state. The quantity of interest is the time-dependent population probability of the CT state, defined as pl(t) = y" a (11) (t~ (2.12) I'~/) 1 , 0 2 , 0 3 ;Vl ,/J2 J)3 ~" TM ~
OI,V2,U3
Keef
=
k=,--K: ]./2,
(2.7)
~(ij)
3 =
where
(t)'~-(
pV 1,02,03;V' I .Ot2.V~ x
(2.8)
The diabatic PE curves of this effective-mode model are shown in Fig. 2. It is seen that the intersection of the S O and the CT potential functions of the effective-mode model occurs close to the
XO I
I( Xll2 I( XO 3
p
I j)
×1 x<> I xo, > Io >. Converged results are obtained with maximum occupation numbers v~'~x= 13, v~ a ' = 12, v~ a ' = 15, The dimension of the relevant Hilbert space of the three-mode model is thus N = 5824. The direct nu-
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113-118
merical solution of the N 2 coupled equations of motion for the R D M would be rather expensive. We employ, therefore, the Monte Carlo wavefunction scheme, see Ref. [33] for more details. Reasonably accurate results are obtained with 1000 propagations of a vector of dimension N, which can be done within a few days of CPU time on a modern workstation. In the single-mode case, the accuracy of the stochastic wavefunction method has been checked by direct numerical propagation of the density matrix. A fourth-order predictor-corrector finite-differences scheme is employed for the wavefunction and density-matrix propagations,
3. Results and discussion Fig. 3 displays the time-dependent population dynamics of the three-mode model with dissipation (full line) in comparison with the population dynamics of the isolated three-mode system (dashed line). It is seen that the three-mode model exhibits an ultrafast electron back-transfer process on a time scale of less than 100 fs which is unaffected by environmental dissipation. In the isolated system, the population of the CT state fluctuates around 0.3 at long times. Upon inclusion of damping, Pz(t) decays to a very small asymptotic value. The additional decay of Pz(t) caused by the coupling to the environment occurs on a time scale of -~ 100 fs, corresponding to an ET rate k = 7 . 7 × l012 S - I . This reflects the fact that high vibrational levels of the electronic ground state are populated in the initial ~0 ,. o o_ 0 . 5
-~ ~.'1
,, ,,, ,,;,.,,5,~ ~ ~, '-l,; ,'
00
. 0.0
. 200,0
.
~ '-~ . . . . ,~ . 400.0
.
lime [fs I
. ,.., '
,"
. 600.0
800.0
Fig. 3. Time-dependent population probability of the CT state of the three-mode model with (full curve) and without (dashed curve) dissipation. Note that the initial ultrafast internal-conversion process is not affected by environmental damping.
lo ~-
~
117
.a ,~, , , ,,". , ,~, .~, ~ .~, .:~ fi '4 ,~. ,l "~ ? ~ .... ,,,,,,, ,' ~. ., ,.' ,. . . -- , ;: .'; ~., ,. . . ' , ,' '~-"~ ,'. , g~'~ ' . ,, ~ "q ~ , l kh . ' , , ~', , "
~ 0..~ o _~ R ~.
°'°00
20~0
40b.0
60b0
~000
time lfs]
Fig. 4. Time-dependent population probability of the CT state of the effective single-mode model with (full curve) and without (dashed curve) dissipation. In the single-mode model, ET is insignificant in the absence of dissipation. The population decay rate obtained with inclusion of dissipation is considerably smaller than in the three-mode model of Fig. 3.
intramolecular internal-conversion process, which possess a shorter lifetime than low vibrational levels. Fig. 3 illustrates the fact that the initial ultrafast CT process is already present in the three-mode system, but is incomplete, because the considerable excess energy can be distributed over only three modes. The coupling to the environment allows dissipation of the excess energy and thus relaxation to the vibronic ground state of the three-mode system. This picture of ultrafast energy deposition into few intramolecular modes, which is followed by vibrational cooling due to the solvent, seems to be in accord with experimental observations for ultrafast ET systems, e.g., S~-S o back-transfer in betaine [4,6,34] or in mixed-valence transition-metal dimers [5,7] in nonpolar or weakly polar solvents. To emphasize the importance of a multi-mode description of the ultrafast intramolecular ET process, we show in Fig. 4 the population dynamics of the effective single-mode model as defined in Eqs. (2.7) and (2.8). The CT population obtained in the absence of dissipation (full line) does not exhibit the ultrafast initial decay seen in Fig. 3, but rather oscillates quasi-periodically around a value of 0.85. The lack of ET in the effective-mode model, despite the favourable PE crossing (cf. Fig. 2) is due to the lack of sufficient density of vibrational states in the final electronic state in the one-dimensional description. Upon inclusion of dissipation, assuming a rate [ ~ = 0.005 as in Fig. 3, an exponential decay of the C T population with an ET rate k = 1.9 × 10 ~2 s- [ is
118
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113-118
obtained (full line in Fig. 4) which is smaller than the damping rate F / h = 7.6 × 1012 s - l of the oscillators. An increase of the damping of the effective mode by a factor of four would be required in order to qualitatively reproduce the population decay of the damped three-mode system in Fig. 3, which would be in conflict with the assumption of weak
system-bath coupling. 4. C o n c l u s i o n We have considered a model of ultrafast ET dynamics which explicitly includes three strongly coupled intramolecular modes as well as dissipation due to weak coupling to an environment. The electronic population dynamics of this model has been evaluated by numerically solving the equation of motion for the RDM, employing the Monte Carlo wavefunction propagation method. The electronic inter-state coupling as well as the intra-state electronvibrational couplings are thus treated in a nonperturbative manner. The time-dependent numerical approach appears to be particularly suitable for the description of ultrafast ET processes in strongly coupied systems, where the assumptions of the conventional golden rule approach are not justified. The model considered in the present work should be adequate for the description of ultrafast electron back-transfer processes in nonpolar solvents, where the main solvent effect is vibrational cooling during and after the internal-conversion process. In polar solvents the model has to be extended to account for the coupling of the electronic charge with the polarization of the medium,
Acknowledgement We would like to thank Luis Seidner for stimulating discussions. This work has been supported by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie.
References [1] R.A. Marcus and N. Sutin, Biochim. Biophys. Acta 811 (1985) 265.
[2] M.D. Newton and N. Sutin, Ann. Rev. Phys. Chem. 35 (1984) 437. [3] H. Heitele, Angew. Chem. Intern. Ed. Engl. 32 (1993) 359. [4] E. ,~kesson, G.C. Walker and P.F. Barbara, J. Chem. Phys. 95 (1991) 4188. [5] P,J. Reid, C. Silva, P.F. Barbara, L. Karki and J.T. Hupp, J. Phys. Chem. 99(1995)2609. [6] E. ,~kesson, A.E. Johnson, N.E. Levinger,G.C. Walker, T.P. DuBruil and P.F. Barbara, J. Chem. Phys. 96 (1992) 7859. [7] S.K. Doom, R.B. Dyer, P.O. Stoutlandand W.H. Woodruff, J. Am. Chem. Soc. 115 (1993) 6398. [8] K.G. Spears, X. Wen and S.M. Arrivo, J. Phys. Chem. 98 (1994) 9693. [9] Y. Nagasawa, A.P. Yartsev, K. Tominaga, P.B. Bisht, A.E. Johnson and K. Yoshihara, J. Phys. Chem. 99 (1995) 653. [10] M.H. Vos, J.-C. Lambry, S.J. Robles, D.G. Youvan, J. Breton and J.-L. Martin, Proc. Natl. Acad. Sci. US 88 (1991) 8885. [11] H. Sumi and R.A. Marcus, J. Chem. Phys. 84 (1986) 4894. [12] J. Jortner and M. Bixon, J. Chem. Phys. 88 (1988) 167. [13] G.C. Walker, E. ,~kesson, A.E. Johnson, N.E. Levinger and P.F. Barbara, J. Phys. Chem. 96 (1992) 3728. [14] M.D. Todd, A. Nitzan and M.A. Rather, J. Phys. Chem. 97 (1993) 29. [151 K.G. Spears, J. Phys. Chem. 99 (1995) 2469. [16] N. Gayathriand B. Bagchi, J. Phys. Chem. 100 (1996) 3056. [17] v. May and M. Schreiber, Phys. Rev. A 45 (1992) 2868. [18] J.M. Jean, R.A. Friesner and G.R. Fleming, J. Chem. Phys. 96 (1992) 5827. [19] v. May, O. Kiihn and M. Schreiber, J. Phys. Chem. 97 (1993) 12591. [20] W.T. Pollard and R.A. Friesner, J. Chem. Phys. 100 (1994) 5054. [21] M. Sugawara, M. Hayashi, S. Suzuki and S.H. Lin, Mol. Phys. 87 (1996) 637. [22] A.G. Redfield, Advan. Magn. Reson. 1 (1965) 1. [23] H. K~ppel,W. Domcke and L.S. Cederbaum, Advan. Chem. Phys. 57 (1984)59. [24] A.D. Hammerich, A. Nitzan and M.A. Ratner, Theoret. Chim. Acta 89 (1994) 383. [25] H. Pl/Shn, S. Krempl, M. Winterstetter and W. Domcke, Chem. Phys. 200 (1995) 11. [26] J. Dalibard, Y. Castin and K. MOlmer, Phys. Rev. Lett. 68 (1992) 580. [27] R. Dum, P. Zoller and H. Ritsch, Phys. Rev. A 45 (1992) 4879. [28] K. Blum, Density matrix theory and applications (Plenum Press, New York, 1981). [29] Y.J. Yan and S. Mukamel, J. Chem. Phys. 89 (1988) 5160. [30] s. Jang, Nucl. Phys. A 499 (1989) 250. [31] Y. Toyozawa and M. Inoue, J. Phys. Soc. Japan 21 (1966) 1663. [32] T. Takagahara, E. Hanamura and R. Kubo, J. Phys. Soc. Japan 44 (1978) 728. [33] B. Wolfseder and W. Domcke, Chem. Phys. Lett. 235 (1995) 370. [34] A.E. Johnson, N.E. Levinger, W. Jarzeba, R.E. Schlief, D.A.V. Klinerand P.F. Barbara, Chem. Phys. 176 (1993)555.
30 August 1996
PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 259 (1996) I 13-118
Intramolecular electron-transfer dynamics in the inverted regime: quantum mechanical multi-mode model including dissipation Brigitte Wolfseder, Wolfgang Domcke Institute of Physical and Theoretical Chemistry, Technical Unioersity of Munich, D-85747 Garching, Germany Received 19 April 1996; in final form 20 June 1996
Abstract
A model of ultrafast electron transfer in condensed media is proposed; it includes electronic inter-state coupling, coupling of the electronic dynamics with several intramolecular modes, and damping of the vibrational motion due to interaction with a thermal environment. Exact numerical solutions of the equation of motion for the reduced density matrix are obtained for this dissipative multi-level system, employing the Monte Carlo wavefunction propagation method. The comparison of the electronic population dynamics of the multi-mode model with an effective single-mode description reveals the importance of multi-mode effects for ultrafast ET dynamics.
1. Introduction Electron-transfer (ET) reactions in the condensed phase have been of interest for decades, see, e.g., Refs. [1-3] for reviews. For many years, attention has been focused on the effect of solvent dynamics on ET reactions. More recently, an increasing number of experiments with femtosecond time resolution are providing evidence for the importance of intramolecular vibrational motion in ET processes, see, e.g., Refs. [4-9]. ET processes in the so-called inverted regime [1], in particular, seem to be controlled more by intramolecular vibrational dynamics rather than solvent effects [4-8]. Moreover, quantum beats reflecting damped coherent vibrational motion have been observed in several ET systems [5,10]. These observations point to the necessity of a fully quan-
tum mechanical and molecule-specific description of ET processes in the femtosecond time regime. Theoretical ET models which include intramolecular vibrational motion have been proposed by Sumi and Marcus [11] and Jortner and Bixon [12]. These early descriptions, which have been further elaborated recently [13-16], are based on the Fermi golden rule expression for the ET rate, and thus assume that vibrational energy and phase relaxation processes are fast compared to the ET reaction. In the case of ET dynamics in the inverted regime, where the reaction can be very fast and the vibrational excess energy is large [5-7], the golden rule description becomes questionable and should be replaced by a nonperturbative treatment. In recent years, the interplay of ET, coherent vibrational motion and dissipation by the solvent has
000%2614/96/$12.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PH S 0 0 0 9 - 2 6 14(96)0071 5-4
114
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113-118
been investigated via the direct numerical solution of the equation of motion for the reduced density matrix (RDM) for models which take into account intramolecular motion as well as coupling to a thermal environment [17-21]. Given the time-dependent RDM, the expectation values of observables pertaining to the system, e.g. electronic state populations or amplitudes of vibrational motion, can straightforwardly be evaluated. This fully quantum mechanical and nonperturbative approach appears to be particularly suitable for the treatment of ET in the inverted regime, where the electronic coupling may be strong and large amount of excess energy has to be dumped into vibrational motion, Since the computational cost of constructing the Redfield relaxation tensor [22] and propagating the RDM scales at least as N 2, where N is the dimension of the relevant Hilbert space of the system, the RDM approach has so far been limited to ET models with a single effective reaction mode [17-21]. It is to
pulses of finite duration, will be considered in a future publication.
be expected, however, that in many cases the ET in donor-acceptor systems involves a change of the equilibrium geometry of several intramolecular modes. A realistic description of the ET reaction may thus require the inclusion of several strongly coupled vibrational modes in the system Hamiltonian [23-25]. In the present work we demonstrate the computational feasibility of a multi-mode description of ET including dissipation by a thermal environment. We employ the so-called Monte Carlo wavefunction propagation method introduced by Dalibard et al., Zoller et al. and others for quantum optical master equations [26,27]. The wavefunction is propagated with a non-Hermitian effective Hamiltonian in the full direct-product Hilbert space of the system, including stochastic quantum jumps [26,27]. Expectation values of system operators are obtained by averaging over a sufficiently large number of realizations of the stochastic process [26,27]. In the present communication we assume, for simplicity and clarity, instantaneous excitation of the charge-transfer (CT)state by an ultrashort laser pulse, The observable of interest is the time-dependent population Pl(t) of the CT state or, equivalently, the ET rate -l~(t)/P~(t). The simulation of real-time pump-probe spectra, taking account of the preparation as well as the probing of the system by laser
Here Et denotes the vertical electronic excitation energy and g is the electronic inter-state coupling. The H i are vibrational Hamiltonians of the respectire electronic states. In terms of creation and annihilation operators pertaining to dimensionless normal modes of the electronic ground state, the H i read (h = 1)
2. Definition of the problem
2.1. Model Hamiltonian We consider a model of a typical photoinduced ET process, where a CT state I q~> is populated by optical excitation from the electronic ground state [ ~P0)- Internal conversion of the excited state to the ground state leads to ultrafast back-transfer of the electron. In adiabatic electronic representation, the system Hamiltonian is written as
Hs ~- E
I ~i)( Hi 4- ~i.l El)( ~°i l
i~ 0,1
+{I ~00)g(,pj 14,h.c.}.
(2.1)
M
Hi = E {wk(btkbk 4" ½) 4" ~i l/£k(bk 4-b~)/~/2"}. k=l
(2.2)
M is the number of vibrational modes to be included in the system Hamiltonian, tok is the vibrational frequency of the kth mode and Ks is the coupling parameter for the intra-state electronic-vibrational coupling, defined as the gradient of the excited-state potential-energy (PE) surface with respect to the kth mode. The dimensionless parameter Kk/to k determines the shift of the equilibrium geometry of the excited state in the kth mode. We shall assume that only few (say, three or four) vibrational modes are strongly coupled to the electronic transition. The remaining weakly or only indirectly coupled modes of the chromophore as well as the solvent degrees of freedom are considered as a thermal bath of harmonic oscillators. Assuming bilinear system-bath coupling and neglecting the ef-
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113 - 118
fect of g on dissipation, the bath variables can be traced out in the standard way, invoking perturbation theory for the system-bath coupling as well as the Markovian approximation [22,28]. The resulting equation of motion for the reduced density operator p(t) of the system reads 0 ~ p ( t ) = - i [ H s, p ( t ) ] + L ( p ) . (2.3) The relaxation operator L(p), expressed in terms of ground-state vibrational creation and annihilation operators, is given by
L ( p ) =½F ~., [~i>A(ij)(q~j],
(2.4)
115
Eq. (2.3) with the system Hamiitonian (2.1) and the relaxation operator (2.4) describes coupled electronic-state dynamics with M strongly coupled systern modes, which are in turn damped by the interaction with a thermal environment. The approximations which have been made in deriving this simple model are justified in the case of weak system-bath coupling and weak to moderate electronic inter-state coupling g. The range of validity of the model defined by Eqs. (2.3)-(2.5) extends, however, well beyond the golden rule regime. In the present application, we include three modes in the system Hamiltonian (M = 3). The vibrational
i,j= 0,1 M
((nk) + 1)(2bk p(O)b~ - b k*b, p(ij)
A(iJ) = E
~ a.o
k=l
\
\
/
~ 1.5 z30
-P(iJ)b~bk)
~ t.o
M
+ E (n,>(2b~
p(iJ)b, - bkb ~ p(ij)
~ o,s
k=l
- p(iJ)b k b~)
o.o
-4.o
M
"4- a i I E ' k=l
Kk
M
Kk
~
2.o
(b -bl)o (").
2.0
(2.5)
k=l
t.o
0.5
I ~:>
0.0
and
(n k) = [exp( WJkBT ) - 1] - '
•
g
Here p(ij) = < ~i I p
o.o
"~(iJ)(bCk -- bk)
~'~'O')k t~
E
'
-2.o
normal coordinate O~
-4.0
-2.0 0.0 normal coordinate Qz
2,0
(2.6)
is the thermal occupation distribution of the kth mode. In the derivation of Eqs. (2.3)-(2.5) the rotating-wave approximation for the system-bath coupiing has been invoked, which is justified for weak system-bath coupling [28]. In the case of strong system-bath coupling, non-secular relaxation terms have to be included. The corresponding relaxation operator, which is equivalent to the classical Brownian oscillator model [29] is also well known [30] and allows the modeling of overdamped vibrational motion. For simplicity, the damping rate F is assumed to be the same for all system modes,
>~ a.o g ~ 2.0 g t,0 o.o
-10.0 -8,0 -6.0 -4.0 -2.0
0.0
normal coordinate Qs
2,0
Fig. 1. Cuts through the S O and S~ diabatic PE surfaces of the three-dimensional ET model along each of the three normal coordinates. For each single mode, the crossing of the diabatic
potentials lies above the verticalexcitationenergy.
116
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113-118
frequencies are taken as w I = 0.26 eV, w 2 = 0.18 eV and to 3 = 0.07 eV. These values are representative for real molecules, as the entire range from low-frequency to high-frequency modes is covered. The electron-vibrational coupling constants are taken as K l 0.34 eV, ~c2 = 0.22 eV and K3 0.09 eV. The dimensionless geometry shift KJ % is thus approximately the same for all three system modes. The vertical energy gap is chosen as E l = 0.91 eV. The parameters have been chosen such that the model describes an ultrafast electron back-transfer process. The inter-state coupling constant is g = 0.06 =
ao ->~ ~.s g t.o
=
eV, and the damping rate of the oscillators is taken as F = 0.005 eV. With this choice of parameters, the approximations made in the derivation of the relaxation operator (2.4), (2.5) are well justified. Since temperature effects are not important for the strongly exothermic ET process considered here, the bath temperature T is chosen to be zero. Fig. 1 shows cuts through the diabatic PE surfaces of the S O and S, states along each of the three normal coordinates. It can be seen that for each mode the crossing of the diabatic surfaces lies above the vertical excitation energy E l . The minimum of the three-dimensional surface of intersection, however, lies very close to the minimum of the CT surface. We thus expect ultrafast surface-crossing dynamics in this three-mode model, In most theoretical models of ET which include intramolecular vibrational dynamics, it has been assumed that the intramolecular vibrational motion can be approximated by a single reaction mode, see, e.g., Refs. [11,12,17-21]. We are able to test the adequacy of this assumption by approximating the three-mode coupled-surface dynamics by an effective single-mode model. Following Refs. [31,32] the coupling constant and frequency of the effective mode are defined as
-~ 0.s O.
0.0 -4,o
-2.0
o.o
normal coordinate Q .
2,0
Fig. 2. Diabatic S O and Sj PE functions of the effective singlemode model. The surface crossing occurs close to the minimum of the S 1 PE function.
minimum of the latter. According to common reasoning, rapid ET is expected in this situation [14,16].
2.2. Numerical methods For tonian sented tronic
the numerical solution of Eq. (2.3) the Hamiland the reduced density operator are reprein the direct-product basis of diabatic elecstates [ ~o0), [ ~Pl) and harmonic-oscillator
states
I x.,)
for each mode
p(t) = ~.~ I ot)p~,a,(a'
l,
(2.9)
",'~' where I a) = I ~i)1X,,)
X,2)I Xo3).
(2.10)
Eq. (2.3) is solved with the initial condition
p(0) = I ~'l ) 10) 10) 10)(0 I(0 I(0 1( ~ I,
(2.11)
which corresponds to vertical excitation from the ground state to the CT state. The quantity of interest is the time-dependent population probability of the CT state, defined as pl(t) = y" a (11) (t~ (2.12) I'~/) 1 , 0 2 , 0 3 ;Vl ,/J2 J)3 ~" TM ~
OI,V2,U3
Keef
=
k=,--K: ]./2,
(2.7)
~(ij)
3 =
where
(t)'~-(
pV 1,02,03;V' I .Ot2.V~ x
(2.8)
The diabatic PE curves of this effective-mode model are shown in Fig. 2. It is seen that the intersection of the S O and the CT potential functions of the effective-mode model occurs close to the
XO I
I( Xll2 I( XO 3
p
I j)
×1 x<> I xo, > Io >. Converged results are obtained with maximum occupation numbers v~'~x= 13, v~ a ' = 12, v~ a ' = 15, The dimension of the relevant Hilbert space of the three-mode model is thus N = 5824. The direct nu-
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113-118
merical solution of the N 2 coupled equations of motion for the R D M would be rather expensive. We employ, therefore, the Monte Carlo wavefunction scheme, see Ref. [33] for more details. Reasonably accurate results are obtained with 1000 propagations of a vector of dimension N, which can be done within a few days of CPU time on a modern workstation. In the single-mode case, the accuracy of the stochastic wavefunction method has been checked by direct numerical propagation of the density matrix. A fourth-order predictor-corrector finite-differences scheme is employed for the wavefunction and density-matrix propagations,
3. Results and discussion Fig. 3 displays the time-dependent population dynamics of the three-mode model with dissipation (full line) in comparison with the population dynamics of the isolated three-mode system (dashed line). It is seen that the three-mode model exhibits an ultrafast electron back-transfer process on a time scale of less than 100 fs which is unaffected by environmental dissipation. In the isolated system, the population of the CT state fluctuates around 0.3 at long times. Upon inclusion of damping, Pz(t) decays to a very small asymptotic value. The additional decay of Pz(t) caused by the coupling to the environment occurs on a time scale of -~ 100 fs, corresponding to an ET rate k = 7 . 7 × l012 S - I . This reflects the fact that high vibrational levels of the electronic ground state are populated in the initial ~0 ,. o o_ 0 . 5
-~ ~.'1
,, ,,, ,,;,.,,5,~ ~ ~, '-l,; ,'
00
. 0.0
. 200,0
.
~ '-~ . . . . ,~ . 400.0
.
lime [fs I
. ,.., '
,"
. 600.0
800.0
Fig. 3. Time-dependent population probability of the CT state of the three-mode model with (full curve) and without (dashed curve) dissipation. Note that the initial ultrafast internal-conversion process is not affected by environmental damping.
lo ~-
~
117
.a ,~, , , ,,". , ,~, .~, ~ .~, .:~ fi '4 ,~. ,l "~ ? ~ .... ,,,,,,, ,' ~. ., ,.' ,. . . -- , ;: .'; ~., ,. . . ' , ,' '~-"~ ,'. , g~'~ ' . ,, ~ "q ~ , l kh . ' , , ~', , "
~ 0..~ o _~ R ~.
°'°00
20~0
40b.0
60b0
~000
time lfs]
Fig. 4. Time-dependent population probability of the CT state of the effective single-mode model with (full curve) and without (dashed curve) dissipation. In the single-mode model, ET is insignificant in the absence of dissipation. The population decay rate obtained with inclusion of dissipation is considerably smaller than in the three-mode model of Fig. 3.
intramolecular internal-conversion process, which possess a shorter lifetime than low vibrational levels. Fig. 3 illustrates the fact that the initial ultrafast CT process is already present in the three-mode system, but is incomplete, because the considerable excess energy can be distributed over only three modes. The coupling to the environment allows dissipation of the excess energy and thus relaxation to the vibronic ground state of the three-mode system. This picture of ultrafast energy deposition into few intramolecular modes, which is followed by vibrational cooling due to the solvent, seems to be in accord with experimental observations for ultrafast ET systems, e.g., S~-S o back-transfer in betaine [4,6,34] or in mixed-valence transition-metal dimers [5,7] in nonpolar or weakly polar solvents. To emphasize the importance of a multi-mode description of the ultrafast intramolecular ET process, we show in Fig. 4 the population dynamics of the effective single-mode model as defined in Eqs. (2.7) and (2.8). The CT population obtained in the absence of dissipation (full line) does not exhibit the ultrafast initial decay seen in Fig. 3, but rather oscillates quasi-periodically around a value of 0.85. The lack of ET in the effective-mode model, despite the favourable PE crossing (cf. Fig. 2) is due to the lack of sufficient density of vibrational states in the final electronic state in the one-dimensional description. Upon inclusion of dissipation, assuming a rate [ ~ = 0.005 as in Fig. 3, an exponential decay of the C T population with an ET rate k = 1.9 × 10 ~2 s- [ is
118
B. Wolfseder, W. Domcke / Chemical Physics Letters 259 (1996) 113-118
obtained (full line in Fig. 4) which is smaller than the damping rate F / h = 7.6 × 1012 s - l of the oscillators. An increase of the damping of the effective mode by a factor of four would be required in order to qualitatively reproduce the population decay of the damped three-mode system in Fig. 3, which would be in conflict with the assumption of weak
system-bath coupling. 4. C o n c l u s i o n We have considered a model of ultrafast ET dynamics which explicitly includes three strongly coupled intramolecular modes as well as dissipation due to weak coupling to an environment. The electronic population dynamics of this model has been evaluated by numerically solving the equation of motion for the RDM, employing the Monte Carlo wavefunction propagation method. The electronic inter-state coupling as well as the intra-state electronvibrational couplings are thus treated in a nonperturbative manner. The time-dependent numerical approach appears to be particularly suitable for the description of ultrafast ET processes in strongly coupied systems, where the assumptions of the conventional golden rule approach are not justified. The model considered in the present work should be adequate for the description of ultrafast electron back-transfer processes in nonpolar solvents, where the main solvent effect is vibrational cooling during and after the internal-conversion process. In polar solvents the model has to be extended to account for the coupling of the electronic charge with the polarization of the medium,
Acknowledgement We would like to thank Luis Seidner for stimulating discussions. This work has been supported by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie.
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