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Electronic decoherence in condensed phases
Chemical Physics 268 (2001) 285±293
www.elsevier.nl/locate/chemphys
Electronic decoherence in condensed phases Daren M. Lockwood, Hyonseok Hwang, Peter J. Rossky * Department of Chemistry and Biochemistry, Institute for Theoretical Chemistry, University of Texas at Austin, Austin, TX 78712-1167, USA Received 12 January 2001
Abstract The impact of electronic decoherence on electronic dynamics, and electron transfer (ET) reactions in particular, in condensed phase environments is discussed. Analytical expressions are considered for a common model system in which ET occurs between relatively displaced harmonic surfaces. We identify a relationship in the semi-classical Marcus limit between electronic decoherence and the Wigner distribution of nuclear con®gurations. Generalization to more complex surfaces is discussed. Additionally, we discuss electronic decoherence for ET via so-called bridge sites in materials, for both direct (superexchange) mechanisms and sequential mechanisms of ET. Quantitative simulation studies of factors aecting characteristic decoherence times for ET in solution are also presented and discussed. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Electron transfer (ET) reactions are central to many biological processes [1] and have been the subject of extensive experimental and theoretical research [2]. However, a number of questions remain unresolved [2]. Because full quantum mechanical treatment of the nuclear degrees of freedom is computationally interactable [3], most theoretical work has addressed systems either in the high temperature limit, where nuclei can be treated classically, or in the low temperature limit, in which case nuclear modes are approximately harmonic [2]. An approach to modeling the important intermediate temperature regime is to represent the
*
Corresponding author. Tel.: +1-512-471-3555; fax: +1-512471-1624. E-mail address: [email protected] (P.J. Rossky).
nuclei by semi-classical (SC) wavepackets. One procedure developed by Heller [4] is based on the fact that Gaussian wavepackets evolving on harmonic potential surfaces preserve their Gaussian shape, with the wavepacket centers following classical trajectories. Heller further suggested that for more complex surfaces, one represent the nuclei by Gaussian wavepackets that evolve according to a locally quadratic approximation to the potential surface. This SC method and other methods derived from it have had appreciable success in treating nuclear dynamics [5,6]. Recently, Neria and Nitzan [3,7] have developed methods for evaluating ET rates based on frozen Gaussian (FG) wavepackets. In this especially convenient FG approach, the wavepacket widths are constant and are determined by the temperature of the system. The Gaussian wavepacket approach to quantum nuclear dynamics provides an attractive means for quantitatively investigating the eect of a
0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 3 0 2 - 0
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D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
condensed phase environment on the electronic dynamics of a system interacting with that environment. Loss of electronic coherence occurs when nuclear wavepacket trajectories on alternative electronic surfaces diverge from one another and is expected to occur very rapidly in condensed phase environments [8,9]. Much progress has already been made toward an understanding of the role played by electronic coherence in general [10± 12], but quantitative treatment of electronic decoherence is required for accurate prediction of electronic dynamics, including ET rates [3,8,13,14]. The evolution of electronic coherence and its dissipation can be explicitly addressed within the FG wavepacket framework [8,13,14], and it is this topic that we address in this paper. The remainder of the paper is organized as follows. In Section 2, we discuss the eect of electronic decoherence within a displaced harmonic surface model for ET, where exact expressions can be obtained. We concentrate on the SC Marcus limit, where the role of electronic decoherence and the relationship with the Wigner distribution of nuclear con®gurations are especially clear. Generalization to more complex surfaces is discussed. In Section 3, we discuss the direct (so-called ``superexchange'') mechanism of ET in proteins, and show that a single decoherence time associated with an eective two-state model is sucient to characterize ET. Decoherence associated with sequential ET is also discussed. In Section 4, we describe the quantitative calculation of characteristic electronic decoherence times. Section 5 presents a summary of our conclusions. 2. Transition between displaced harmonic surfaces We begin by considering two electronic surfaces which are harmonic in the nuclear coordinates and dier only by displacement of the equilibrium nuclear positions. Much literature on ET reactions has been devoted to consideration of this displaced harmonic oscillator (HO) model, due to the fact that exact analytical rate expressions can be obtained [3,7,15,16]. This model is most clearly applicable to condensed phase ET at low temperatures where nuclear modes are approximately
harmonic, but allows for incorporation of power spectra based on ambient atomistic simulation, as in the ``dispersed polaron'' model of Warshel and Hwang [16]. Here, we examine the role electronic coherence plays in determining the ET rate within this displaced HO model. The Hamiltonian is H j DiHD h Dj j AiHA h Aj V j Aih Dj j Dih Aj 1 where j Di denotes the initial (``donor'') electronic state, and j Ai the ®nal (``acceptor'') electronic state. The electronic coupling is denoted by V, while HD and HA are the nuclear Hamiltonians on the electronic surfaces j Di and j Ai respectively HD
N 1X 2 p2 =mn mn x2n xn xn0 2 n1 n
N 1X HA p2 =mn mn x2n xn 2 n1 n
xn0
2
2 2hx0
Here, the index n denotes each nuclear coordinate, with frequency xn , coordinate displacement between the two states speci®ed by xn0 , and a D±A energy splitting speci®ed by x0 . The rate of transition from a state j Dijii, where jii denotes the initial nuclear state, to the electronic surface j Ai is given in the case of small electronic coupling by the golden rule [2,3,7]. We initially consider V to be static for simplicity, and discuss generalization below. The golden rule rate is given by k V =h
2
Z
1 1
dt C t
3
where the correlation function C t is C t hijeiHA t=h e
iHD t=h
jii
4
If this correlation function is expressed in the position representation, and a thermal average over initial nuclear states jii is taken, the resulting expression for the thermal correlation function hC tiT is [15]
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
Z hC tiT
dxN
(
2 xN i T exp ix0 t
X
mn xn xn0 = h2ixn sin xn t
n
) 2
xn0 1 Z
cos xn t coth b hxn =2
(
2 N dx wi x T exp ix0 t
n
Z )
2i= hmn x2n xn xn0 t # t3
N
where x represents a nuclear con®guration x1 ; x2 ; . . . ; xn , and the thermal average over initial
nuclear wave functions w2i xN T can be identi®ed at high temperatures as the classical probability distribution of nuclear coordinates. In condensed phase environments, the correlation function decays rapidly, and higher order terms in time in the exponent can commonly be neglected [3]. Smaller frequencies xn also favor this limit [15]. The rate can then be written as Z 1 Z
2 ksc V = h dt dxN w2i xN T 1 ( ) X 2 exp ix0 t 2i= hmn xn xn xn0 t
2
Z
2pV =h
1 1
Z
n
Z dt
dx
N
w2i xN
T
exp fiHAD t= hg
dxN w2i xN T d HAD
exp f# b5 g dxN p 4 x1 ; T p 4 x2 ; T . . . p 4 xN ; T exp fiHAD t=hg exp f# b5 g
5
V = h 2
In the displaced HO model considered
here, the analytical (Wigner) form of w2i xN T is well known [7,15]. Here we consider the high temperature result. The thermal correlation function is then Z 4 hCsc tiT dxN pN xN ; T exp fiHAD t=hg
N
X
287
7
where the superscript (4) denotes neglect of terms of order higher than b4 , and p 4 xn ; T is given by 2
p 4 xn ; T Nn 1 expf bmn x2n xn xn0 =2g 2
exp fh2 b3 mn x4n xn xn0 =24g
8
where Nn is a normalization constant. To this order in time, Eq. (7) can be written in a form more convenient for evaluation [17]
2 4 hCsc tiT exp fihHAD i 4 t=hg expf 1=2 DHAD t2 =h2 # t4 g ( ) X exp ix0 t 2i mn x2n x2n0 t=h exp
n
n
o
2 4 2 2 1=2 DHAD t =h # t4
6
9
where HAD HA HD is dierence in potential energy between the surfaces. Eq. (6) is often referred to as the SC Marcus expression. In this limit, ET can be viewed as only occurring at nuclear con®gurations where the potential energy surfaces cross, and the ET rate is proportional to the thermal probability of attaining a nuclear con®guration located at a point of intersection. The neglected higher order terms in time may be interpreted as a source of electronic transitions in regions where the potential surfaces do not cross [15].
where DHAD HAD hHAD i 4 , and h i 4 denotes averaging over nuclear con®gurations weighted by 4 pN x; T . A generalization of Eq. (9) to more complex surfaces than the harmonic case is suggested by the work of Neria and Nitzan [3,7], in which FG wavepackets are used to represent nuclei. We ®nd that p 4 xn ; T can be obtained in the harmonic case by convolution of the classical nuclear probability distribution p 2 xn ; T with Gaussian functions based on the usual [3,8,14] high temperature thermal nuclear width kn h= 6mn kT 1=2 . That is,
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D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
p 4 xn ; T /
Z
dx0n p 2 x0n ; T exp x0n
(
2
xn
6mn kT = h2 # b5
2
10
More general expressions for the nuclear width can be derived [3,7] that are expected to give good agreement with the Wigner distribution in other temperature domains. Here, however, we concentrate on the high temperature case. Because this high temperature wavepacket width does not depend on the surface curvature, a straightforward generalization of Eqs. (9) and (10) might be expected to perform well for potential surfaces more complex than the harmonic case. That is, a reasonable explicit approach would be the formation of p 4 xn ; T by convolution of the same Gaussian function, explicit in Eq. (10), with a classical nuclear distribution that is appropriate for the general surface of interest. We note the appealing property that evaluation of rates in this approach requires no dynamical nuclear information. Eq. (9) can alternatively be derived from a dynamical perspective, thus making a connection to electronic decoherence [8,18]. To obtain this relation, one assumes that the nuclear state can be represented by FG wavepackets following classical trajectories. Then, evaluation of the thermal average of Eq. (4) gives [8,18] Z hCsc tiT
2
dxN pN xN ; T D t exp iHAD t= h 11
where the decoherence function D t is included explicitly. D t describes the loss of nuclear overlap as wavepackets on the dierent electronic surfaces diverge from one another. It can be readily evaluated to the required order in time based on the dierences in forces on the potential energy surfaces DFn [8], yielding ln D t
t2 =4 h2
X n
k2n DFn2
12
The result for the thermal average correlation function in Eq. (11) is given by
hCsc tiT exp fihHAD i t= hg exp t2 =2 h2 ) " # X 2 2 2 2 4 2 4 hDHAD i 2 kn mn xn xn0 # t n ( exp fihHAD i 4 t= hg exp t2 = h2 # " ) X 2 2 4 2 2 2 4 2mn xn xn0 =b kn mn xn xn0 # t n
13 which agrees with Eq. (9) for the appropriate 1=2 thermal width kn h= 6mn kT to order b4 . Hence, in the SC Marcus limit, what appears in the position representation to be a quantum correction to the transition state population can alternatively be treated as quantum electronic decoherence associated with nuclear wavepackets which classically populate the transition state. The term decoherence is used here to avoid confusion with pure dephasing [5,13]. Pure dephasing is associated only with energy gap ¯uctuations, and like decoherence, leads to decay of o-diagonal elements of the electronic (reduced) density matrix. However, it is notable that pure dephasing rates increase with temperature due to augmentation of energy gap ¯uctuations, and need not depend on the quantum character of the nuclear bath, while decoherence rates decrease with increasing temperature, vanishing in the classical limit. The decoherence function can, in fact, be regarded as a correction factor to the high temperature correlation function, a factor which becomes increasingly important as the temperature is lowered. To make this explicit, we consider the high temperature expansion of the exact quantum correlation function [15,16]. This expansion is considered in more detail elsewhere [19]. One obtains X ln hC tiT ix0 t 2mn xn x2n0 =hfi sin xn t n
1 cos xn t coth bhxn =2g X 2mn xn x2n0 =h ix0 t n
fi sin xn t 1
cos xn t
2=hxn b bhxn =6 # b2 g 14
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
The ®rst correction factor (of order b) becomes more important as the temperature is lowered and is seen to be equal to the decoherence function D t (to order t2 ) using the high temperature thermal wavepacket width. We have seen in our previous work that under ambient conditions, it is often imperative that decoherence be taken into account [8,14]. The rate of decay of the correlation function due to decoherence is expected to be remarkably fast (on the fs time scale) [8,14], providing a justi®cation for neglecting the higher order terms in time in the correlation function. It should also be noted that, in the case of a high rate of coherence loss, the ¯uctuations in the electronic coupling evaluated along the classical trajectory can be slow enough that the square of an eective coupling can be taken outside of the integral over time. In this case, the ``eective'' static coupling corresponds to the value for those con®gurations dominant in the SC limit, namely, those where the dierence in potential energy between the surfaces is zero [20]. It is important to note that Eq. (11) is applicable in cases where the SC Marcus limit is not valid. The only requirements for the validity of Eq. (11) are that, ®rst, representation of the nuclei by FG wavepackets remains valid, and second, the electronic decoherence time is short [8,18]. (Generalization beyond the harmonic case corresponds to generalization of the classical p 2 xn ; T only.) We have focused here on the (less general) SC Marcus limit because the conditions under which decoherence is both necessary and sucient to correct the ``classical'' bath result are especially clear in this case. Speci®cally, it is evident that at high enough temperatures, the decoherence function goes to unity, and the high temperature SC Marcus rate is obtained. But as the temperature is lowered, it becomes necessary to multiply the correlation function by the non-trivial correction factor equal to the decoherence function D t. Additionally, we have shown that in the SC Marcus limit, decoherence associated with nuclear wavepackets can alternatively be treated as an effective change in the thermal transition state population in a position representation. However, the wavepacket picture is of more general utility, due to its straightforward generalization to treat
289
cases where electronic transitions are important for nuclear con®gurations other than the transition state. In particular, we note that a surface hopping method which incorporates electronic decoherence has been valuable in explaining electronic transition rates in H2 O and D2 O solvents [8].
3. Superexchange and electron transfer in proteins In another work [14], we have considered the case of an electronic transition between donor (D) and acceptor (A) sites in a protein, where each electronic state is described by a localized tightly bound distribution. In this form, the formulation used elsewhere to describe (localized) electronic transitions of the hydrated electron [8] is immediately applicable. More generally, however, idealized localized initial and ®nal electronic states can be coupled to the electronic structure of the medium, particularly in a partially structured environment such as an ET protein. To make clear the eect of decoherence on such a so-called superexchange mechanism of ET, we begin by modifying the Hamiltonian of the previous section to include a ``bridging'' (B) electronic state, as follows H H0 H 0 j DiHD h Dj j BiHB h Bj j AiHA h Aj VD j Dih Bj j Bih Dj VA j Aih Bj j Bih Aj 15 where j Bi denotes a so-called bridging electronic state to which the donor and acceptor states are each coupled, and in contrast to Eq. (1), direct coupling between donor and acceptor states is taken to be negligible. Generalization of the Hamiltonian for further bridge sites is straightforward [10]. We no longer take the surfaces to be harmonic, but we do take the coupling to be static for convenience. If the initial state of the system is taken to correspond to a thermal equilibrium of nuclear states on the electronic surface j Di, then elements of the electronic (reduced) density matrix at a time t later are given by [21]
290
qI;J
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
X
N 1e
bED
hEI jh I je
iHt= h
j DijED i
EI ;ED
hED jh DjeiHt=h j J ijEI i
16
where jEI i denotes a stationary nuclear state on the electronic surface j I i labeled by the energy of the state j I ijEI i, and N is a normalization constant. The diagonal elements therefore evolve according to X 2 qI;I N 1 e bED hEI jh I je iHt=h j DijED i 17 EI ;ED
Expansion of the time evolution operator to second order in the coupling gives [21] ( X 2 qI;I N 1 e bED dD;I i=hh I jH 0 j DihEI jED i EI ;ED
Z 0
t
dt0 e
i EI ED t0 = h
X 1=i hEXD EX
0
0
h I jH j X ih X jH j DihEI jEX ihEX jED i ) Z t h i 2 0 0 dt0 e iEID t =h e iEIX t =h
18
0
where EXD EX ED . In particular, the population of the surface j Ai grows according to ( X X 2 1 bED qA;A N e 1=i hEBD VA VD hEA jEB i EA ;ED
hEB jED i
EB
Z
t 0
0
dt e
)2 iEAD t0 = h
e
iEAB t0 = h
19
0
If the energy gap EBD between ground vibrational states of dierent electronic states is large compared to the energy variation between important 0 vibrational states, the approximation EBD EBD can be made within the brackets in Eq. (19). Then the closure relation can be used to eliminate jEB ihEB j, and the ®rst of the two terms in Eq. (19) has the same form as it would for direct coupling, except that the eective (superexchange) electronic 0 0 coupling is VA VD =EBD . This condition of large EBD
is precisely the condition under which superexchange is the dominant mechanism of ET [10]. Since in this superexchange case only the eective electronic coupling is aected, a single decoherence time based on D and A states is sucient to characterize ET, as in a true two state case. On the other hand, as the energy gap is lowered, there is an increasing rate of transfer to the bridge electronic state, according to 2
qB;B
X EB ;ED
N 1e Z
t 0
bED
0
dt e
i=hVD hEB jED i i EB ED t0 = h
2 20
For times longer than the characteristic B±D decoherence time, the o-diagonal terms involving the bridge will have decayed to zero. The rate of transfer to the acceptor state will then scale with the bridge population, according to a sequential (incoherent) mechanism [10]. That is, ET occurs ®rst to the bridge and then second from the bridge to acceptor state, in which case, there are separate relevant decoherence times for each step. It is interesting to note [10] that ecient decoherence is a requirement for eective sequential transfer (``incoherent'' mechanism). The wavepacket picture provides a compelling way to visualize the superexchange and sequential transfer mechanisms in the SC limit. In this picture, transfer to the bridge or acceptor surfaces occurs where these surfaces intersect the donor surface, but in the case of bridge-mediated ET, an eective (superexchange) electronic coupling determines the crossing rate at the transition state [20]. Decoherence aects the crossing rate, and also determines the period of time after which interference eects may be neglected (i.e., when there can be considered to be simply a statistical mixture of electronic states). The eective static electronic coupling is, as noted above, to be evaluated at the curve crossing [20]. Of course, the rate of thermal equilibration of nuclei on the bridge surface aects the viability of expressing the sequential transfer rate simply in terms of two thermal rates, one for each of the two transfer steps.
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
4. Quantitative calculations In this section, we discuss calculation of a characteristic electronic decoherence time associated with direct ET between metal ions. For the case of direct state-to-state electronic transitions, including direct (superexchange) ET between tightly bound electronic donor and acceptor states, the electronic state can be taken to be a linear combination of the initial electronic state j1i and the ®nal electronic state j2i throughout its evolution. The electronic transition rate can often be obtained via the golden rule [2,3,7]. Classical molecular dynamics or Monte Carlo simulations can be used to obtain an ensemble of dierent nuclear wavepacket con®gurations on the initial electronic surface. Then a characteristic (thermally averaged) decoherence time can be evaluated via Eq. (12) using the high temperature wavepacket widths and an appropriate classical force ®eld [8,14]. In earlier work, we have considered the hydrated electron in detail [8]. In that work, molecular dynamics simulation was used, employing a fully molecular description of the solvent and an eective electron±molecule pseudopotential to describe the excess electron. The transition considered primarily was the relaxation of the ®rst excited state of the electron to the ground state, which has the character of a 2p to 1s transition in a hydrogen atom. For this transition in H2 O, which involves a change in charge distribution but no change in average dipole moment, the electronic decoherence time is found to be about 3 fs [8]. The very small value can be attributed to the additive contributions (see Eq. (12)) from the large number of high frequency modes in the system, associated with the large density of (low mass) protons in water. We have also considered the characteristic electronic decoherence time for ET between metal centers in ruthenated Pseudomonas aeruginosa azurin [14], which is a well-characterized electron-transport protein [17,22,23]. This protein contains a single copper redox center and a surface histidine group to which ruthenium complexes may be attached, as described by Gray and Winkler [22,24]. These complexes act as electron donor and ac-
291
ceptor, and permit photoinduced ET over controlled distances in natural structures. A fully atomistic simulation of Ru(bpy)2 (im)-(His 83)-modi®ed azurin [24] in 6500 water molecules was performed [14]. To calculate the dierences in forces on the nuclei corresponding to each electronic state, we considered not only the eect of charge redistribution accompanying ET, but also the change in covalent radii of the metal ions. In this case, electronic decoherence also occurs remarkably quickly, characteristically in about 2.4 fs, with both solvation dynamics and protein dynamics contributing. This rapid electronic decoherence notably favors the SC limit, as discussed earlier. As noted above, earlier work on the hydrated electron indicated that the water contribution to decoherence is dominated by (polar) hydrogenic contributions. Further, one should expect that the electrostatic character of the forces should lead to long ranged contributions. To see the range and magnitude quantitatively in a simpler system, we consider the electronic decoherence rate associated with ET between a pair of ions comin water, for a ®xed ion separation of 17 A, mensurate with the separation of the metal ions in the protein. The ions are modeled here as Na and Cl , for simplicity, and the water is modeled via the SPC model [25]. The result is shown in Fig. 1. In the ®gure, the decoherence time (see Eq. (12)) is shown that would be obtained if only contributions from solvent molecules inside a spherical cuto distance from each ion were included. The results in the ®gure clearly demonstrate that water is an exceptionally eective medium in inducing electronic coherence loss for the charge transfer case, reaching a decoherence time of only 1 fs when all solvent is included. Further, it is clear that ``all'' solvent extends in a practical sense to although the ®rst solvation shells of about 20 A, is sucient to the ions alone (inside about 5 A) reduce the coherence time to under 2 fs. The potential ability to make practical estimates of decoherence times based on simple electrostatics and simple model systems is of obvious interest. Such a simple estimate is also shown in Fig. 1. The result shown is for a model calculation in which charge transfer occurs between hard spheres of
292
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
Fig. 1. Contributions to sD 2 for ET between ions in a pair in water. The value is given cumulatively as a separated by 17 A function of the maximum distance between a contributing solvent molecular center and an ion center. The diamond symbols denote the contribution for the fully atomistic simulation; the solid line denotes the result obtained from the mean density model described in the text.
beyond which there is a uniform radius 2.0 A, distribution of protons, each with the same partial charge as in the model water. The contribution from oxygen atoms is included by simply scaling the proton density by a factor of 1.0625 times, in accord with the fact that each proton will typically contribute eight times as much as one oxygen (see Eq. (12)). For this model, the spatial distribution of force contributions entering the expression for the decoherence time is analytical, so that the result shown involves only a simple integral. The agreement between the rough estimate and the full calculation is remarkably good, indicating that the origin of the decoherence rate in such cases of charge transfer are really quite straightforward to understand and to quantitatively estimate.
5. Conclusions We have considered the dynamics of electronic state change for an electronic system interacting with its environment. Quantum nuclear eects in the environment have been discussed from various perspectives. The appearance of these eects in the
form of electronic decoherence has been identi®ed analytically for a two-level system in a harmonic nuclear bath in the SC high temperature case. The further equivalence in this case to the use of a quantum Wigner distribution of nuclear coordinates in evaluation of thermal rate constants from the Golden Rule has been established. The use of these alternative wavepacket-based forms as corrections to classical results for general non-harmonic surfaces is suggested, as is implicit in earlier implementations of decoherence in surface hopping simulations [8]. We have also discussed the more general case of coupling of the states in the two-level system to an intermediate ``bridge'' state, as appears in the socalled superexchange mechanism of ET. The conditions for the usefulness in this case of the same two-state analysis were shown not to be restrictive, and the role of decoherence rate in establishing the regime for superexchange or sequential ET mechanisms was discussed. Quantitative calculations of decoherence rates based on the SC wavepacket expressions were also outlined. For charge transfer in water, it was shown explicitly that the decoherence time is extremely fast (1 fs), comparable to the ultrafast decoherence rates seen in earlier work on the hydrated electron excited state relaxation and ET in a hydrated protein. It was shown that these rates can be understood simply based on the electrostatic interactions with the polar atoms of the environment, and that a corresponding consequence was a long range of interactions contributing to electronic decoherence. These analytical and numerical results clarify the relationship among alternative approaches to quantum rate calculations and suggest new practical approximations. It is expected that such quantum-corrected classical bath descriptions based on the decoherence framework will provide an increasingly useful conceptual and numerical approach to complex material systems. Acknowledgements The authors would like to thank Valeri Barsegov for stimulating discussions. The authors are
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
grateful to the Robert A. Welch Foundation and the NSF for support of the research reported here.
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www.elsevier.nl/locate/chemphys
Electronic decoherence in condensed phases Daren M. Lockwood, Hyonseok Hwang, Peter J. Rossky * Department of Chemistry and Biochemistry, Institute for Theoretical Chemistry, University of Texas at Austin, Austin, TX 78712-1167, USA Received 12 January 2001
Abstract The impact of electronic decoherence on electronic dynamics, and electron transfer (ET) reactions in particular, in condensed phase environments is discussed. Analytical expressions are considered for a common model system in which ET occurs between relatively displaced harmonic surfaces. We identify a relationship in the semi-classical Marcus limit between electronic decoherence and the Wigner distribution of nuclear con®gurations. Generalization to more complex surfaces is discussed. Additionally, we discuss electronic decoherence for ET via so-called bridge sites in materials, for both direct (superexchange) mechanisms and sequential mechanisms of ET. Quantitative simulation studies of factors aecting characteristic decoherence times for ET in solution are also presented and discussed. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Electron transfer (ET) reactions are central to many biological processes [1] and have been the subject of extensive experimental and theoretical research [2]. However, a number of questions remain unresolved [2]. Because full quantum mechanical treatment of the nuclear degrees of freedom is computationally interactable [3], most theoretical work has addressed systems either in the high temperature limit, where nuclei can be treated classically, or in the low temperature limit, in which case nuclear modes are approximately harmonic [2]. An approach to modeling the important intermediate temperature regime is to represent the
*
Corresponding author. Tel.: +1-512-471-3555; fax: +1-512471-1624. E-mail address: [email protected] (P.J. Rossky).
nuclei by semi-classical (SC) wavepackets. One procedure developed by Heller [4] is based on the fact that Gaussian wavepackets evolving on harmonic potential surfaces preserve their Gaussian shape, with the wavepacket centers following classical trajectories. Heller further suggested that for more complex surfaces, one represent the nuclei by Gaussian wavepackets that evolve according to a locally quadratic approximation to the potential surface. This SC method and other methods derived from it have had appreciable success in treating nuclear dynamics [5,6]. Recently, Neria and Nitzan [3,7] have developed methods for evaluating ET rates based on frozen Gaussian (FG) wavepackets. In this especially convenient FG approach, the wavepacket widths are constant and are determined by the temperature of the system. The Gaussian wavepacket approach to quantum nuclear dynamics provides an attractive means for quantitatively investigating the eect of a
0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 3 0 2 - 0
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D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
condensed phase environment on the electronic dynamics of a system interacting with that environment. Loss of electronic coherence occurs when nuclear wavepacket trajectories on alternative electronic surfaces diverge from one another and is expected to occur very rapidly in condensed phase environments [8,9]. Much progress has already been made toward an understanding of the role played by electronic coherence in general [10± 12], but quantitative treatment of electronic decoherence is required for accurate prediction of electronic dynamics, including ET rates [3,8,13,14]. The evolution of electronic coherence and its dissipation can be explicitly addressed within the FG wavepacket framework [8,13,14], and it is this topic that we address in this paper. The remainder of the paper is organized as follows. In Section 2, we discuss the eect of electronic decoherence within a displaced harmonic surface model for ET, where exact expressions can be obtained. We concentrate on the SC Marcus limit, where the role of electronic decoherence and the relationship with the Wigner distribution of nuclear con®gurations are especially clear. Generalization to more complex surfaces is discussed. In Section 3, we discuss the direct (so-called ``superexchange'') mechanism of ET in proteins, and show that a single decoherence time associated with an eective two-state model is sucient to characterize ET. Decoherence associated with sequential ET is also discussed. In Section 4, we describe the quantitative calculation of characteristic electronic decoherence times. Section 5 presents a summary of our conclusions. 2. Transition between displaced harmonic surfaces We begin by considering two electronic surfaces which are harmonic in the nuclear coordinates and dier only by displacement of the equilibrium nuclear positions. Much literature on ET reactions has been devoted to consideration of this displaced harmonic oscillator (HO) model, due to the fact that exact analytical rate expressions can be obtained [3,7,15,16]. This model is most clearly applicable to condensed phase ET at low temperatures where nuclear modes are approximately
harmonic, but allows for incorporation of power spectra based on ambient atomistic simulation, as in the ``dispersed polaron'' model of Warshel and Hwang [16]. Here, we examine the role electronic coherence plays in determining the ET rate within this displaced HO model. The Hamiltonian is H j DiHD h Dj j AiHA h Aj V j Aih Dj j Dih Aj 1 where j Di denotes the initial (``donor'') electronic state, and j Ai the ®nal (``acceptor'') electronic state. The electronic coupling is denoted by V, while HD and HA are the nuclear Hamiltonians on the electronic surfaces j Di and j Ai respectively HD
N 1X 2 p2 =mn mn x2n xn xn0 2 n1 n
N 1X HA p2 =mn mn x2n xn 2 n1 n
xn0
2
2 2hx0
Here, the index n denotes each nuclear coordinate, with frequency xn , coordinate displacement between the two states speci®ed by xn0 , and a D±A energy splitting speci®ed by x0 . The rate of transition from a state j Dijii, where jii denotes the initial nuclear state, to the electronic surface j Ai is given in the case of small electronic coupling by the golden rule [2,3,7]. We initially consider V to be static for simplicity, and discuss generalization below. The golden rule rate is given by k V =h
2
Z
1 1
dt C t
3
where the correlation function C t is C t hijeiHA t=h e
iHD t=h
jii
4
If this correlation function is expressed in the position representation, and a thermal average over initial nuclear states jii is taken, the resulting expression for the thermal correlation function hC tiT is [15]
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
Z hC tiT
dxN
(
2 xN i T exp ix0 t
X
mn xn xn0 = h2ixn sin xn t
n
) 2
xn0 1 Z
cos xn t coth b hxn =2
(
2 N dx wi x T exp ix0 t
n
Z )
2i= hmn x2n xn xn0 t # t3
N
where x represents a nuclear con®guration x1 ; x2 ; . . . ; xn , and the thermal average over initial
nuclear wave functions w2i xN T can be identi®ed at high temperatures as the classical probability distribution of nuclear coordinates. In condensed phase environments, the correlation function decays rapidly, and higher order terms in time in the exponent can commonly be neglected [3]. Smaller frequencies xn also favor this limit [15]. The rate can then be written as Z 1 Z
2 ksc V = h dt dxN w2i xN T 1 ( ) X 2 exp ix0 t 2i= hmn xn xn xn0 t
2
Z
2pV =h
1 1
Z
n
Z dt
dx
N
w2i xN
T
exp fiHAD t= hg
dxN w2i xN T d HAD
exp f# b5 g dxN p 4 x1 ; T p 4 x2 ; T . . . p 4 xN ; T exp fiHAD t=hg exp f# b5 g
5
V = h 2
In the displaced HO model considered
here, the analytical (Wigner) form of w2i xN T is well known [7,15]. Here we consider the high temperature result. The thermal correlation function is then Z 4 hCsc tiT dxN pN xN ; T exp fiHAD t=hg
N
X
287
7
where the superscript (4) denotes neglect of terms of order higher than b4 , and p 4 xn ; T is given by 2
p 4 xn ; T Nn 1 expf bmn x2n xn xn0 =2g 2
exp fh2 b3 mn x4n xn xn0 =24g
8
where Nn is a normalization constant. To this order in time, Eq. (7) can be written in a form more convenient for evaluation [17]
2 4 hCsc tiT exp fihHAD i 4 t=hg expf 1=2 DHAD t2 =h2 # t4 g ( ) X exp ix0 t 2i mn x2n x2n0 t=h exp
n
n
o
2 4 2 2 1=2 DHAD t =h # t4
6
9
where HAD HA HD is dierence in potential energy between the surfaces. Eq. (6) is often referred to as the SC Marcus expression. In this limit, ET can be viewed as only occurring at nuclear con®gurations where the potential energy surfaces cross, and the ET rate is proportional to the thermal probability of attaining a nuclear con®guration located at a point of intersection. The neglected higher order terms in time may be interpreted as a source of electronic transitions in regions where the potential surfaces do not cross [15].
where DHAD HAD hHAD i 4 , and h i 4 denotes averaging over nuclear con®gurations weighted by 4 pN x; T . A generalization of Eq. (9) to more complex surfaces than the harmonic case is suggested by the work of Neria and Nitzan [3,7], in which FG wavepackets are used to represent nuclei. We ®nd that p 4 xn ; T can be obtained in the harmonic case by convolution of the classical nuclear probability distribution p 2 xn ; T with Gaussian functions based on the usual [3,8,14] high temperature thermal nuclear width kn h= 6mn kT 1=2 . That is,
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D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
p 4 xn ; T /
Z
dx0n p 2 x0n ; T exp x0n
(
2
xn
6mn kT = h2 # b5
2
10
More general expressions for the nuclear width can be derived [3,7] that are expected to give good agreement with the Wigner distribution in other temperature domains. Here, however, we concentrate on the high temperature case. Because this high temperature wavepacket width does not depend on the surface curvature, a straightforward generalization of Eqs. (9) and (10) might be expected to perform well for potential surfaces more complex than the harmonic case. That is, a reasonable explicit approach would be the formation of p 4 xn ; T by convolution of the same Gaussian function, explicit in Eq. (10), with a classical nuclear distribution that is appropriate for the general surface of interest. We note the appealing property that evaluation of rates in this approach requires no dynamical nuclear information. Eq. (9) can alternatively be derived from a dynamical perspective, thus making a connection to electronic decoherence [8,18]. To obtain this relation, one assumes that the nuclear state can be represented by FG wavepackets following classical trajectories. Then, evaluation of the thermal average of Eq. (4) gives [8,18] Z hCsc tiT
2
dxN pN xN ; T D t exp iHAD t= h 11
where the decoherence function D t is included explicitly. D t describes the loss of nuclear overlap as wavepackets on the dierent electronic surfaces diverge from one another. It can be readily evaluated to the required order in time based on the dierences in forces on the potential energy surfaces DFn [8], yielding ln D t
t2 =4 h2
X n
k2n DFn2
12
The result for the thermal average correlation function in Eq. (11) is given by
hCsc tiT exp fihHAD i t= hg exp t2 =2 h2 ) " # X 2 2 2 2 4 2 4 hDHAD i 2 kn mn xn xn0 # t n ( exp fihHAD i 4 t= hg exp t2 = h2 # " ) X 2 2 4 2 2 2 4 2mn xn xn0 =b kn mn xn xn0 # t n
13 which agrees with Eq. (9) for the appropriate 1=2 thermal width kn h= 6mn kT to order b4 . Hence, in the SC Marcus limit, what appears in the position representation to be a quantum correction to the transition state population can alternatively be treated as quantum electronic decoherence associated with nuclear wavepackets which classically populate the transition state. The term decoherence is used here to avoid confusion with pure dephasing [5,13]. Pure dephasing is associated only with energy gap ¯uctuations, and like decoherence, leads to decay of o-diagonal elements of the electronic (reduced) density matrix. However, it is notable that pure dephasing rates increase with temperature due to augmentation of energy gap ¯uctuations, and need not depend on the quantum character of the nuclear bath, while decoherence rates decrease with increasing temperature, vanishing in the classical limit. The decoherence function can, in fact, be regarded as a correction factor to the high temperature correlation function, a factor which becomes increasingly important as the temperature is lowered. To make this explicit, we consider the high temperature expansion of the exact quantum correlation function [15,16]. This expansion is considered in more detail elsewhere [19]. One obtains X ln hC tiT ix0 t 2mn xn x2n0 =hfi sin xn t n
1 cos xn t coth bhxn =2g X 2mn xn x2n0 =h ix0 t n
fi sin xn t 1
cos xn t
2=hxn b bhxn =6 # b2 g 14
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
The ®rst correction factor (of order b) becomes more important as the temperature is lowered and is seen to be equal to the decoherence function D t (to order t2 ) using the high temperature thermal wavepacket width. We have seen in our previous work that under ambient conditions, it is often imperative that decoherence be taken into account [8,14]. The rate of decay of the correlation function due to decoherence is expected to be remarkably fast (on the fs time scale) [8,14], providing a justi®cation for neglecting the higher order terms in time in the correlation function. It should also be noted that, in the case of a high rate of coherence loss, the ¯uctuations in the electronic coupling evaluated along the classical trajectory can be slow enough that the square of an eective coupling can be taken outside of the integral over time. In this case, the ``eective'' static coupling corresponds to the value for those con®gurations dominant in the SC limit, namely, those where the dierence in potential energy between the surfaces is zero [20]. It is important to note that Eq. (11) is applicable in cases where the SC Marcus limit is not valid. The only requirements for the validity of Eq. (11) are that, ®rst, representation of the nuclei by FG wavepackets remains valid, and second, the electronic decoherence time is short [8,18]. (Generalization beyond the harmonic case corresponds to generalization of the classical p 2 xn ; T only.) We have focused here on the (less general) SC Marcus limit because the conditions under which decoherence is both necessary and sucient to correct the ``classical'' bath result are especially clear in this case. Speci®cally, it is evident that at high enough temperatures, the decoherence function goes to unity, and the high temperature SC Marcus rate is obtained. But as the temperature is lowered, it becomes necessary to multiply the correlation function by the non-trivial correction factor equal to the decoherence function D t. Additionally, we have shown that in the SC Marcus limit, decoherence associated with nuclear wavepackets can alternatively be treated as an effective change in the thermal transition state population in a position representation. However, the wavepacket picture is of more general utility, due to its straightforward generalization to treat
289
cases where electronic transitions are important for nuclear con®gurations other than the transition state. In particular, we note that a surface hopping method which incorporates electronic decoherence has been valuable in explaining electronic transition rates in H2 O and D2 O solvents [8].
3. Superexchange and electron transfer in proteins In another work [14], we have considered the case of an electronic transition between donor (D) and acceptor (A) sites in a protein, where each electronic state is described by a localized tightly bound distribution. In this form, the formulation used elsewhere to describe (localized) electronic transitions of the hydrated electron [8] is immediately applicable. More generally, however, idealized localized initial and ®nal electronic states can be coupled to the electronic structure of the medium, particularly in a partially structured environment such as an ET protein. To make clear the eect of decoherence on such a so-called superexchange mechanism of ET, we begin by modifying the Hamiltonian of the previous section to include a ``bridging'' (B) electronic state, as follows H H0 H 0 j DiHD h Dj j BiHB h Bj j AiHA h Aj VD j Dih Bj j Bih Dj VA j Aih Bj j Bih Aj 15 where j Bi denotes a so-called bridging electronic state to which the donor and acceptor states are each coupled, and in contrast to Eq. (1), direct coupling between donor and acceptor states is taken to be negligible. Generalization of the Hamiltonian for further bridge sites is straightforward [10]. We no longer take the surfaces to be harmonic, but we do take the coupling to be static for convenience. If the initial state of the system is taken to correspond to a thermal equilibrium of nuclear states on the electronic surface j Di, then elements of the electronic (reduced) density matrix at a time t later are given by [21]
290
qI;J
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
X
N 1e
bED
hEI jh I je
iHt= h
j DijED i
EI ;ED
hED jh DjeiHt=h j J ijEI i
16
where jEI i denotes a stationary nuclear state on the electronic surface j I i labeled by the energy of the state j I ijEI i, and N is a normalization constant. The diagonal elements therefore evolve according to X 2 qI;I N 1 e bED hEI jh I je iHt=h j DijED i 17 EI ;ED
Expansion of the time evolution operator to second order in the coupling gives [21] ( X 2 qI;I N 1 e bED dD;I i=hh I jH 0 j DihEI jED i EI ;ED
Z 0
t
dt0 e
i EI ED t0 = h
X 1=i hEXD EX
0
0
h I jH j X ih X jH j DihEI jEX ihEX jED i ) Z t h i 2 0 0 dt0 e iEID t =h e iEIX t =h
18
0
where EXD EX ED . In particular, the population of the surface j Ai grows according to ( X X 2 1 bED qA;A N e 1=i hEBD VA VD hEA jEB i EA ;ED
hEB jED i
EB
Z
t 0
0
dt e
)2 iEAD t0 = h
e
iEAB t0 = h
19
0
If the energy gap EBD between ground vibrational states of dierent electronic states is large compared to the energy variation between important 0 vibrational states, the approximation EBD EBD can be made within the brackets in Eq. (19). Then the closure relation can be used to eliminate jEB ihEB j, and the ®rst of the two terms in Eq. (19) has the same form as it would for direct coupling, except that the eective (superexchange) electronic 0 0 coupling is VA VD =EBD . This condition of large EBD
is precisely the condition under which superexchange is the dominant mechanism of ET [10]. Since in this superexchange case only the eective electronic coupling is aected, a single decoherence time based on D and A states is sucient to characterize ET, as in a true two state case. On the other hand, as the energy gap is lowered, there is an increasing rate of transfer to the bridge electronic state, according to 2
qB;B
X EB ;ED
N 1e Z
t 0
bED
0
dt e
i=hVD hEB jED i i EB ED t0 = h
2 20
For times longer than the characteristic B±D decoherence time, the o-diagonal terms involving the bridge will have decayed to zero. The rate of transfer to the acceptor state will then scale with the bridge population, according to a sequential (incoherent) mechanism [10]. That is, ET occurs ®rst to the bridge and then second from the bridge to acceptor state, in which case, there are separate relevant decoherence times for each step. It is interesting to note [10] that ecient decoherence is a requirement for eective sequential transfer (``incoherent'' mechanism). The wavepacket picture provides a compelling way to visualize the superexchange and sequential transfer mechanisms in the SC limit. In this picture, transfer to the bridge or acceptor surfaces occurs where these surfaces intersect the donor surface, but in the case of bridge-mediated ET, an eective (superexchange) electronic coupling determines the crossing rate at the transition state [20]. Decoherence aects the crossing rate, and also determines the period of time after which interference eects may be neglected (i.e., when there can be considered to be simply a statistical mixture of electronic states). The eective static electronic coupling is, as noted above, to be evaluated at the curve crossing [20]. Of course, the rate of thermal equilibration of nuclei on the bridge surface aects the viability of expressing the sequential transfer rate simply in terms of two thermal rates, one for each of the two transfer steps.
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
4. Quantitative calculations In this section, we discuss calculation of a characteristic electronic decoherence time associated with direct ET between metal ions. For the case of direct state-to-state electronic transitions, including direct (superexchange) ET between tightly bound electronic donor and acceptor states, the electronic state can be taken to be a linear combination of the initial electronic state j1i and the ®nal electronic state j2i throughout its evolution. The electronic transition rate can often be obtained via the golden rule [2,3,7]. Classical molecular dynamics or Monte Carlo simulations can be used to obtain an ensemble of dierent nuclear wavepacket con®gurations on the initial electronic surface. Then a characteristic (thermally averaged) decoherence time can be evaluated via Eq. (12) using the high temperature wavepacket widths and an appropriate classical force ®eld [8,14]. In earlier work, we have considered the hydrated electron in detail [8]. In that work, molecular dynamics simulation was used, employing a fully molecular description of the solvent and an eective electron±molecule pseudopotential to describe the excess electron. The transition considered primarily was the relaxation of the ®rst excited state of the electron to the ground state, which has the character of a 2p to 1s transition in a hydrogen atom. For this transition in H2 O, which involves a change in charge distribution but no change in average dipole moment, the electronic decoherence time is found to be about 3 fs [8]. The very small value can be attributed to the additive contributions (see Eq. (12)) from the large number of high frequency modes in the system, associated with the large density of (low mass) protons in water. We have also considered the characteristic electronic decoherence time for ET between metal centers in ruthenated Pseudomonas aeruginosa azurin [14], which is a well-characterized electron-transport protein [17,22,23]. This protein contains a single copper redox center and a surface histidine group to which ruthenium complexes may be attached, as described by Gray and Winkler [22,24]. These complexes act as electron donor and ac-
291
ceptor, and permit photoinduced ET over controlled distances in natural structures. A fully atomistic simulation of Ru(bpy)2 (im)-(His 83)-modi®ed azurin [24] in 6500 water molecules was performed [14]. To calculate the dierences in forces on the nuclei corresponding to each electronic state, we considered not only the eect of charge redistribution accompanying ET, but also the change in covalent radii of the metal ions. In this case, electronic decoherence also occurs remarkably quickly, characteristically in about 2.4 fs, with both solvation dynamics and protein dynamics contributing. This rapid electronic decoherence notably favors the SC limit, as discussed earlier. As noted above, earlier work on the hydrated electron indicated that the water contribution to decoherence is dominated by (polar) hydrogenic contributions. Further, one should expect that the electrostatic character of the forces should lead to long ranged contributions. To see the range and magnitude quantitatively in a simpler system, we consider the electronic decoherence rate associated with ET between a pair of ions comin water, for a ®xed ion separation of 17 A, mensurate with the separation of the metal ions in the protein. The ions are modeled here as Na and Cl , for simplicity, and the water is modeled via the SPC model [25]. The result is shown in Fig. 1. In the ®gure, the decoherence time (see Eq. (12)) is shown that would be obtained if only contributions from solvent molecules inside a spherical cuto distance from each ion were included. The results in the ®gure clearly demonstrate that water is an exceptionally eective medium in inducing electronic coherence loss for the charge transfer case, reaching a decoherence time of only 1 fs when all solvent is included. Further, it is clear that ``all'' solvent extends in a practical sense to although the ®rst solvation shells of about 20 A, is sucient to the ions alone (inside about 5 A) reduce the coherence time to under 2 fs. The potential ability to make practical estimates of decoherence times based on simple electrostatics and simple model systems is of obvious interest. Such a simple estimate is also shown in Fig. 1. The result shown is for a model calculation in which charge transfer occurs between hard spheres of
292
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
Fig. 1. Contributions to sD 2 for ET between ions in a pair in water. The value is given cumulatively as a separated by 17 A function of the maximum distance between a contributing solvent molecular center and an ion center. The diamond symbols denote the contribution for the fully atomistic simulation; the solid line denotes the result obtained from the mean density model described in the text.
beyond which there is a uniform radius 2.0 A, distribution of protons, each with the same partial charge as in the model water. The contribution from oxygen atoms is included by simply scaling the proton density by a factor of 1.0625 times, in accord with the fact that each proton will typically contribute eight times as much as one oxygen (see Eq. (12)). For this model, the spatial distribution of force contributions entering the expression for the decoherence time is analytical, so that the result shown involves only a simple integral. The agreement between the rough estimate and the full calculation is remarkably good, indicating that the origin of the decoherence rate in such cases of charge transfer are really quite straightforward to understand and to quantitatively estimate.
5. Conclusions We have considered the dynamics of electronic state change for an electronic system interacting with its environment. Quantum nuclear eects in the environment have been discussed from various perspectives. The appearance of these eects in the
form of electronic decoherence has been identi®ed analytically for a two-level system in a harmonic nuclear bath in the SC high temperature case. The further equivalence in this case to the use of a quantum Wigner distribution of nuclear coordinates in evaluation of thermal rate constants from the Golden Rule has been established. The use of these alternative wavepacket-based forms as corrections to classical results for general non-harmonic surfaces is suggested, as is implicit in earlier implementations of decoherence in surface hopping simulations [8]. We have also discussed the more general case of coupling of the states in the two-level system to an intermediate ``bridge'' state, as appears in the socalled superexchange mechanism of ET. The conditions for the usefulness in this case of the same two-state analysis were shown not to be restrictive, and the role of decoherence rate in establishing the regime for superexchange or sequential ET mechanisms was discussed. Quantitative calculations of decoherence rates based on the SC wavepacket expressions were also outlined. For charge transfer in water, it was shown explicitly that the decoherence time is extremely fast (1 fs), comparable to the ultrafast decoherence rates seen in earlier work on the hydrated electron excited state relaxation and ET in a hydrated protein. It was shown that these rates can be understood simply based on the electrostatic interactions with the polar atoms of the environment, and that a corresponding consequence was a long range of interactions contributing to electronic decoherence. These analytical and numerical results clarify the relationship among alternative approaches to quantum rate calculations and suggest new practical approximations. It is expected that such quantum-corrected classical bath descriptions based on the decoherence framework will provide an increasingly useful conceptual and numerical approach to complex material systems. Acknowledgements The authors would like to thank Valeri Barsegov for stimulating discussions. The authors are
D.M. Lockwood et al. / Chemical Physics 268 (2001) 285±293
grateful to the Robert A. Welch Foundation and the NSF for support of the research reported here.
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