Dependence of Photosynthetic Electron-Transfer Kinetics on

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10474

J. Phys. Chem. B 2004, 108, 10474-10483

Dependence of Photosynthetic Electron-Transfer Kinetics on Temperature and Energy in a Density-Matrix Model† William W. Parson*,§ and Arieh Warshel*,‡ Department of Biochemistry, UniVersity of Washington, Box 357350, Seattle, Washington 98195-7350 ReceiVed: January 29, 2004; In Final Form: March 18, 2004

We use a multidimensional density-matrix model to explore the temperature dependence of electron transfer from the excited singlet state of the primary electron donor (P*) to the neighboring bacteriochlorophyll (BA) in photosynthetic bacterial reaction centers. This reaction, which has the unusual property of increasing in rate with decreasing temperature, occurs too rapidly to be treated reliably by approaches that assume thermal equilibration of the vibrational levels of the reactant electronic state. In the density-matrix treatment, the frequencies and displacements of the vibrational modes that are coupled to electron transfer, and the microscopic time constants for transitions between different vibrational states, are obtained from molecular-dynamics simulations by the dispersed-polaron (spin-boson) approach. The electron-transfer dynamics are simulated by integrating the stochastic Liouville equation following excitation of the system with a short pulse of light. In this model, the increase in the rate with decreasing temperature depends strongly on the fact that electron transfer occurs more rapidly than vibrational thermalization. The high rate of electron transfer also affects the dependence of the kinetics on the energy difference between the reactant and product electronic states (∆E°), making the optimal value of -∆E° smaller than the reorganization energy. This interesting effect can be rationalized qualitatively in simple semiclassical and quantum mechanical models.

Introduction One of the most striking features of the initial electron-transfer processes that occur in photosynthetic bacterial reaction centers is that the reactions increase in rate with decreasing temperature. This unusual temperature dependence is seen in electron transfer from the excited singlet state of the bacteriochlorophyll dimer (P*) to the nearby bacteriopheophytin (HA),1-6 in the individual steps that underlie this reaction (electron transfer from P* to 7,8 and in the bacteriochlorophyll BA and from BA to HA), transfer of an electron from HA to quinone QA.9-13 The rate constant for electron transfer from P* to HA in reaction centers of Rhodobacter sphaeroides appears to be limited by the initial electron transfer to BA and increases from about 0.3 × 1012 s-1 at room temperature to about 0.5 × 1012 s-1 at 80 K and 0.7 × 1012 s-1 at 20 K.2,7 An increase in rate with decreasing temperature also occurs in the return of an electron from the reduced quinone (QA ) to the oxidized bacteriochlorophyll dimer (P+).14-19 Under some conditions, return of an electron from the secondary quinone (QB ) and electron transfer from a bound cytochrome to P+ also can occur at cryogenic temperatures.20-24 Most of the theoretical discussions of the temperature dependence of these reactions have begun with the assumption that the vibrational levels of the reactant electronic state equilibrate rapidly relative to the rate of electron transfer. Marcus25,26 noted that, with this assumption, the rate should †

Part of the special issue “Gerald Small Festschrift”. * Corresponding authors. Phone: (206) 543-1743 (W.W.P.); (213) 7404114 (A.W.). E-mail: [email protected] (W.W.P.); warshel@ usc.edu (A.W.). § University of Washington. ‡ Department of Chemistry, University of Southern California, Los Angeles, CA 90007.

increase with decreasing temperature if the free energy change in the reaction (∆G°) is approximately equal, but opposite in sign, to the reorganization energy (λ). The rate constant in the semiclassical Marcus theory is

kET )

2π |V |2(4πλkBT)-1/2 exp{-(∆G° + λ)2/4λkBT} (1) p el

where Vel is the electronic coupling matrix element, T is the temperature, and kB is the Boltzmann constant. If ∆G° ) -λ, the activation energy is zero and the rate varies approximately as T-1/2. Because eq 1 neglects nuclear tunneling, it can overestimate the importance of the exponential factor when T is small or ∆G° < -λ.27 However, treatments that include nuclear tunneling have given a qualitatively similar picture: the rate should increase with decreasing temperature if the potential energy surfaces of the reactant and product intersect near the zeropoint level of the reactant state.27-37 Following work by Lax,38 Kubo and Toyozawa,39 and Levich,40 Jortner and coworkers31-33,41,42 showed that, if electron transfer is strongly coupled to a harmonic vibrational mode with energy pω, the rate constant can be written as

kET )

2π |V |2(2πΛ)-1/2 exp{-(∆Eo + λ)2/2Λ} p el

(2)

Here ∆E° is the energy change in the reaction, which is the same as ∆G° if the vibrational frequency is the same in the reactant and product states and effects of configurational entropy are neglected; λ is the quantum reorganization energy, (1/2)pω∆2, where ∆ is the dimensionless displacement of the vibrational potential surface in the product state relative to the reactant; and Λ ) pωλ coth(pω/2kBT). If ∆E° ) -λ, eq 2 reduces to kET ) ko{tanh (pω/2kBT)}1/2 with ko ) (2π/p)|Vel|2(2πλpω)-1/2.

10.1021/jp0495904 CCC: $27.50 © 2004 American Chemical Society Published on Web 05/12/2004

Photosynthetic Electron-Transfer Kinetics The rate of electron transfer from P* to HA in Rb. sphaeroides can be fit well by this expression with pω ) 80 cm-1.2,32 The reorganization energy λ in eq 2 usually is taken to include only the low-frequency modes of the solvent, with high-frequency molecular modes of the electron carriers being treated separately.27,29,43,44 Equation 2, like the Marcus equation, assumes that vibrational levels of the reactant state thermalize rapidly relative to electron transfer. Both treatments also assume that the electron-transfer reaction is nonadiabatic so that the rate is proportional to |Vel|2. More rigorous expressions have been obtained by the dispersed-polaron (spin-boson) treatment, in which moleculardynamics (MD) simulations are used to evaluate the complete spectral distribution of the vibrational modes that are coupled to electron transfer.27,37,45-52 Given the frequencies and displacements for all the pertinent vibrational modes, the rate constant can be obtained by the exact quantum expression39 for a quasiharmonic multimode system. This treatment becomes exact in the harmonic approximation, but still assumes that vibrational thermalization occurs rapidly relative to electron transfer. Although functionally important relaxations of the reaction center are known to occur on time scales of 10-8 s and longer,19,53-59 the assumption of rapid vibrational equilibration seems safe for relatively slow reactions such as charge recombination of P+ QA . But it clearly does not hold for electron transfer from P* to BA. Oscillations seen in the spectra of stimulated emission and excited-state absorption of P* indicate that vibrational coherences can persist for several picoseconds after reaction centers are excited with a short flash,60-73 and the evolution of the spectra depends on the excitation wavelength.74 Oscillations in the absorption bands of BA suggest that electron transfer begins in a stepwise manner, with a small amount of P+ BA forming each time a vibrational wavepacket moving on the P* potential surface crosses the surface for P+ BA , with the dynamics becoming smoother as the wavepacket dephases.66,70-73 Vibrational relaxations could affect the temperature dependence of charge separation in a variety of ways. Several authors have suggested that vibrational relaxations that trap the reaction center in the P+ BA state limit the overall rate of charge separation.55,56,70-73,75 Rapid vibrational dephasing, on the other hand, could impede electron transfer by quenching off-diagonal 76 elements of the density matrix that connect P* with P+ BA. In either case, the dependence of the kinetics on temperature could reflect the temperature dependence of vibrational relaxations, rather then the thermal weighting of the Franck-Condon factors for electron transfer from different vibrational substates of P*, as eqs 1 and 2 assume. In addition, the usual Boltzmann weighting factors clearly are inapplicable if electron transfer precedes vibrational thermalization.4,74 Haffa et al.4 have suggested that reactions from hot vibrational states allow electron transfer to occur rapidly at cryogenic temperatures even if the energy of P+ BA is increased by site-directed mutagenesis. Stimulated emission from P* is modulated by a large family of vibrational modes with energies ranging from below 30 to at least 200 cm-1.60-73,77-81 Because the dipole moment of the P+ BA ion-pair state is greater than that of P*, one would expect electron transfer from P* to BA also to be coupled to multiple vibrational modes, and molecular-dynamics (MD) simulations using the dispersed-polaron (spin-boson) formalism are in accord with this expectation.27,36,37,52,76,82-85 A quantum mechanical treatment of the temperature dependence of electron transfer must, therefore, consider coupling to multiple vibrational modes.

J. Phys. Chem. B, Vol. 108, No. 29, 2004 10475 We recently described a multidimensional, density-matrix treatment that is applicable to electron transfer when vibrational relaxations and dephasing occur on similar time scales.76 The density-matrix model includes a set of quantized, harmonic vibrational modes, whose frequencies and displacements are obtained from MD simulations by the dispersed-polaron method. The electron-transfer dynamics are obtained by integrating the stochastic Liouville equation

∂G i ) [G,H ] + R ‚ G ∂t p

(3a)

or

∂Fjk ∂t

)

i p

∑u (Fju ‚ Huk - Hju ‚ Fuk) + ∑u ∑V Rjk,uV ‚ FuV

(3b)

where Fjk, Hjk, and Rij,kl are, respectively, elements of the vibronic density matrix (G), the Hamiltonian interaction matrix (H) for electron transfer, and the Redfield relaxation tensor (R). Interaction matrix element Hjk is a product of the multidimensional Franck-Condon factor for transitions between vibronic states j and k and an electronic coupling factor (Vel) that we take to be the same for all vibrational states. The relaxation tensor R describes stochastic transitions, dephasing, and coherence transfer among different vibrational states of the same electronic state (P* or P+ BA ) under the influence of the solvent: Rjj,kk is the microscopic rate constant for transitions from vibrational state k to state j, -Rkk,kk is the sum of the rate constants for transitions from state k to all the other states, -Rjk,jk is the rate constant for decay of coherence between states j and k, and Rij,kl governs transfer of coherence from one pair of states (k and l) to another (i and j). These rate constants are obtained by requiring the time dependence of the energy gap between P* and P+ BA in the density-matrix model to reproduce the autocorrelation function of the energy gap in the MD simulations. We showed that this can be done reasonably well by adjusting a single parameter

T01 ) (R00,11 + R11,00)-1

(4)

where R00,11 and R11,00 are the rate constants for transitions between the first two levels (zero and one phonon) of an individual vibrational mode.76 T01 is the time constant for equilibration of these levels in the absence of electron transfer. R00,11 and R11,00 are related to each other by R11,00 ) exp(-pωm/kBT)R00,11, where pωm is the energy of the vibrational mode. The rate constants for transitions, dephasing, or coherence transfer between any other pair of vibrational states depend straightforwardly on T01, the numbers of phonons of each mode in the two states, the temperature, and the vibrational energies. The model thus remains computationally tractable while avoiding the use of a single phenomenological time constant (T2) to describe all the interactions of the system with the bath. In principle, the H matrix provides a complete microscopic treatment of the effects of the bath on vibronic transitions between different electronic states. We therefore do not need to describe these transitions by a phenomenological relaxation matrix or time constant. The relaxation tensor R is used only to describe stochastic processes within a given electronic state. The role of the thermal bath in this model is, not to provide a continuum of additional low-frequency vibrational modes for which coupling to electron transfer can be treated classically,

10476 J. Phys. Chem. B, Vol. 108, No. 29, 2004 but rather to allow transitions between different vibrational states of the quantum system. Our previous study76 examined the effects of varying T01, the electronic coupling between P* and P+ BA , the energy difference between the two electronic states (∆E°), the reorganization energy, and other parameters, and included a preliminary survey of the effects of temperature. We found that values of T01 in the range of 1 to 2 ps gave the energy gap appropriate dynamics and led to stepwise electron-transfer kinetics that qualitatively reproduced the key experimental observations on reaction centers. Decreasing T01 to 0.1 ps slowed electron transfer, probably in part because of the loss of hot vibrational levels that favor electron transfer, and in part due to the rapid disappearance of off-diagonal density-matrix elements that connect the reactant and product electronic states. In the present work, we use the density-matrix model to explore the dependence of the electron-transfer kinetics on temperature and energy in detail. We find that the effects of both temperature and ∆E° hinge strongly on the rate of vibrational equilibration.

Parson and Warshel A trajectory in the density-matrix model begins with impulsive excitation of P to P*. All the calculations described here assume an excitation pulse with a Gaussian spectral profile centered 150 cm-1 above the lowest vibrational level of P*. The vibrational overlap integrals for P f P* are calculated from the vibrational frequencies and displacements by standard expressions86 and are used to construct the initial reduced density matrix (G(0)). The expression for G(0) in our previous paper76 (eq 27) is incorrect and gives erroneously large initial values for some of the off-diagonal matrix elements. The desired expressions, expanded to consider a Boltzmann distribution of vibrational levels in the ground state, are

Fjk(0) ) N-1Z-1 5

pi,j,k )

The density-matrix model was essentially as described previously,76 and included five effective vibrational modes with energies (pωm) of 30, 70, 125, 175, and 235 cm-1. The dimensionless displacements of the vibrational coordinates for excitation of P to P* were taken to be 2.5, 2.0, 1.0, 0.5, and 0.2, respectively. The displacements for P* f P+ BA (∆m) were 3.7, 2.0, 1.4, 0.8, and 0.5, giving a total reorganization energy (λ ) 1/2∑mpωm∆m2) of 553 cm-1 for the electron-transfer step. The frequencies and displacements of the vibrations coupled to electron transfer were obtained, as described,37,76 from the Fourier transform of the autocorrelation function of the energy gap between P* and P+ BA , which was recorded during 1-ns molecular-dynamics simulations of the Rb. sphaeroides reaction center. These quantized vibrational modes include motions of the protein and, in principle, represent the complete spectral density of the fluctuations of the energy gap between P and P* for the excitation step and the gap between P* and 76 However, to keep the P+ B A for the electron-transfer step. size of the density matrix of the quantum system within manageable bounds, we divided the spectral density function into the five effective vibrational modes described above. Because the calculated dynamics depend to some extent on the details of this division, they should be viewed primarily as illustrative. Higher-frequency vibrational modes that are seen in the molecular-dynamics simulations37 but are not included in the model would increase the total reorganization energy for P* f P+ BA by about 10%. The vibrational displacements for excitation of P to P* have not been studied extensively by molecular-dynamics simulations, and the values used here are relatively arbitrary. For most of the calculations, we limited the number of states further by cutting off the vibrational levels of P* at a maximum of 15 phonons of any individual mode or a total vibrational energy of 375 cm-1 above the zero-point level. The vibrational levels of P+ BA were cut off at the same absolute energy (e.g., 675 cm-1 above the zero-point level of + -1 below P*). With P* P+ BA when P BA was 300 cm -1 truncated at 375 cm and the zero-point level of P+ BA 300 cm-1 below that of P* (∆E° ) -300 cm-1), the model included 103 vibrational levels of P* and 659 levels of P+ BA . Lowering the cutoff energies by 25 cm-1 decreased the vibrational levels of P* to 84 and those of P+ BA to 580.

5 j i { 〈χm|χm〉}{ 〈χkm|χim〉} × m)1 m)1 exp{-[(j - ex)2 + (k

∏

N ) Z-1

Methods

Z)

exp(-i/kBT)pi,j,k ∑ i∈P

(5a)

∏

exp(-i/kBT) ∑ pi,j,j ∑ i∈P j∈P*

exp(-i/kBT) ∑ i∈P

- ex)2]/2Γex} (5b) (5c) (5d)

Here i denotes a vibronic substate of the ground electronic state (P), and j and k denote substates of P*; χlm is the harmonicoscillator wave function for mode m in vibronic state l; l is the energy of state l, ex is the center of the excitation spectrum; and Γex ) Wex2/4ln2, where Wex (150 cm-1 in the present study) is the full spectral width of the excitation spectrum at halfmaximal amplitude. Equations 5a-d were used for all the present calculations that considered excitation of P from a distribution of vibrational levels. Most of the calculations in which only the lowest vibrational level of P (i ) 0) was populated before the excitation flash were done using eq 27 of the previous paper. However, test calculations showed that, for the situations considered here, the electron-transfer rates obtained with the two treatments of G(0) were not significantly different. Using eqs 5a-d decreases the high-frequency oscillations of the energy gap between P* and P+ BA at short times, but as discussed previously, these oscillations have little effect on the mean electron-transfer rate.76 The time dependence of G after the initial excitation was evaluated by Runge-Kutta integration of the stochastic Liouville equation.76 As indicated above, the Hamiltonian matrix element for electron-transfer transitions between a given pair of vibronic sublevels of P* and P+ BA was calculated from an assumed electronic coupling factor (Vel) and the product of the pertinent overlap integrals for all the vibrational modes. The elements of the Redfield relaxation matrix (Rij,kl) depend on the numbers of phonons of each mode in the vibronic states (i, j, k, and l), and on the time constant for equilibration of the two lowest levels of a vibrational mode (T01). Transfer of coherence between different pairs of density-matrix elements was included only for P*. Except where stated otherwise, the electronic coupling factor Vel was taken to be 10 cm-1 and was assumed to be independent of temperature and to be the same for all vibrational states. P+ BA was assumed to decay from any of its vibrational levels to P+ HA with a time constant of 1 ps, and the time constant for pure dephasing of the two lowest levels of a vibrational mode was set at 20 ps. The effects of varying these parameters

Photosynthetic Electron-Transfer Kinetics

Figure 1. Temperature dependence of the effective rate constant for electron transfer from P* to BA, as expressed by the decrease in the P* population between 0 and 1 ps after coherent excitation, for three values of the energy difference between the potential energy minima of P* -1 -1 and P+ BA (∆E° ) -100 cm , squares; -200 cm , triangles; and -300 cm-1, circles) and T01 ) 1 ps (A) or 0.1 ps (B). T01 was taken to be independent of temperature, and P was assumed to be exclusively in its lowest vibrational state before the excitation. -1 were considered previously.76 P+ HA was placed 1000 cm below P*, making electron transfer from BA to HA essentially irreversible at the temperatures considered here. Other parameters were varied as described below. For the purposes of the present work, we define the effective rate constant for electron transfer as -∆P*/∆t, where ∆P* is the decrease in the sum of the populations of the vibronic substates of P* between 0 and 1 ps and ∆t ) 1 ps. Because electron transfer begins in a stepwise manner if the system is excited coherently, the effective rate constant depends on the choice of ∆t as well as Vel; however, the definition used here gives approximately the same value if the system is excited incoherently.76 To follow vibrational thermalization in the absence of electron transfer, we recorded the excess vibrational energy of P* relative to the vibrational energy for a system in a Boltzmann equilibrium at the same temperature. The thermalization time constant (τe) is defined as the time required for the excess vibrational energy to fall to 1/e of its initial level. The equilibration dynamics are strictly exponential only at very low temperatures.

Results We examined how the rate of electron transfer in the densitymatrix model depends on temperature under several different assumptions about T01, the time constant for vibrational equilibration of the two lowest levels of a vibrational mode. To separate effects of temperature on vibrational equilibration in P* and P+ BA from effects on the initial distribution of vibrational levels in the ground state (P), suppose first that P is entirely in its lowest vibrational state before the excitation. Figure 1 shows the rates calculated for this situation when T01 was taken to be independent of temperature. In Figure 1A, T01 was fixed at 1 ps, which was found previously to make the damping of oscillatory features in energy gap between P* and P + BA at 80 K similar to the damping in dispersed-polaron simulations for the same temperature.76 In Figure 1B, T01 was decreased to 0.1 ps. Considering the longer relaxation time first -1 below P* (∆E° ) (Figure 1A), when P+ BA is put 100 cm -100 cm-1) the calculated rate is almost independent of temperature. With more negative values of ∆E°, the rate increases modestly with decreasing temperature down to about 20 K but drops slightly between 20 and 10 K.

J. Phys. Chem. B, Vol. 108, No. 29, 2004 10477 Decreasing T01 to 0.1 ps reduces the electron-transfer rate and changes the temperature dependence: the rate constant now drops abruptly with decreasing temperature below 60 K, in disagreement with the experimental observations (Figure 1B). This qualitative feature is not very sensitive to the choice of ∆E°. The electron-transfer rate calculated for the present model at 80 K is relatively insensitive to changes in T01 between 0.5 and 2 ps, increasing by only about 5% if T01 is lengthened from 1 to 2 ps and decreasing by about 8% at 0.5 ps. However, it drops rapidly if T01 is reduced below 0.5 ps.76 Two main factors probably contribute to the decrease in the rate when T01 is reduced.76 First, vibrational relaxation in P* competes with electron transfer from hot vibrational states that are populated transiently by the excitation pulse. Second, rapid vibrational transitions destroy the off-diagonal density-matrix elements that are essential intermediates in the flow of population from P* to P+ BA . This is the “quantum Zeno paradox” or “watchedpot” effect, which can be viewed as a result of ascertaining the state of the quantum system at very short intervals of time. The watched-pot effect also can be seen in the density-matrix model as a decrease in the electron-transfer rate if the off-diagonal elements of G are quenched more rapidly by either pure 76 Opposing dephasing or electron transfer from BA to HA. 0 these adverse effects, decreasing T1 or increasing the rate of electron transfer to HA also facilitates escape of P+ BA from resonance with P*, which in itself would increase the overall rate of electron transfer. The watched-pot effect, loss of electron transfer from hot vibrational states, and relaxations of P+ BA all become increasingly important at low temperatures. At high temperatures, thermalization occurs as a diffusive walk through a large number of vibrational states that differ in energy by less than kBT. At low temperatures, it becomes a more direct process. This is illustrated by the filled and empty circles in Figure 2A, which show the time required for the excess vibrational energy to decrease to 1/e of its initial value in the absence of electron transfer (τe). If T01 is fixed at 1 ps, τe decreases from 2.11 ps at 80 K to 1.0 ps at 5 K; with T01 ) 0.1 ps, the corresponding times are 0.211 and 0.10 ps. In addition to accelerating the thermalization dynamics, reducing the temperature has the usual effect of restricting the vibrational states that are accessible in the equilibrated system. The diamonds in Figure 2 are for a more realistic model in which P occupies a distribution of vibrational levels before the excitation, which we consider later. Figure 3A shows the effect of temperature on the damping of vibrational coherences, as reflected by oscillations of the energy gap between P+ BA and P*. Transfer of coherence between different pairs of vibrational states allows these oscillations to persist for times considerably beyond T01. 76,87 However, if T01 is held fixed at 1 ps, low-frequency oscillations that continue for more than 10 ps at 80 K (dotted curve in Figure 3A) are essentially gone by 6 ps at 5 K (solid curve). If T01 is increased by a factor of 2.11 for the trajectory at 5 K, so as to make τe the same here as at 80 K, the low-frequency oscillations of the energy gap become almost indistinguishable at the two temperatures (Figure 3B). Differences in the damping of highfrequency oscillations at the two temperatures are still noticeable, but these are less important in the model considered here because electron transfer is coupled most strongly to the low-frequency modes. To test the significance of the variation of τe with temperature when T01 is held fixed, we examined the temperature depen-

10478 J. Phys. Chem. B, Vol. 108, No. 29, 2004

Figure 2. (A) Temperature dependence of the time (τe) required for the excess vibrational energy of P* to decay to 1/e of its initial value following coherent excitation of P in the absence of electron transfer, with T01 ) 1 ps (filled circles and diamonds) or 0.1 ps (open circles). In the calculations shown with filled and open circles, P was assumed to be in its lowest vibrational state before the excitation and the model included all vibrational levels with energies up to 350 cm-1 above the zero-point levels of P and P*; in those shown with diamonds, P was excited from a Boltzmann distribution of vibrational states and the cutoff energies were 375 cm-1. (Raising the cutoff energies from 350 to 375 cm-1 increased τe by about 4% at 120 K and less at lower temperatures.) The horizontal dashed and dotted lines indicate the default values of τe used for the calculations shown in Figures 4 and 6. (B) The values of T01 used in Figures 4, 6, and 7 to keep τe ) 2.11 ps.

dence of the electron-transfer rate when T01 was adjusted so as to keep τe constant at either 2.11 or 0.211 ps. These calculations were done with ∆E° ) -300 cm-1. Figure 4A shows the results. With τe held at 2.11 ps (its value at 80 K when T01 ) 1 ps), the temperature dependence of the electron-transfer rate is similar to that seen when T01 is held fixed at 1 ps, except that the rate continues to increase slightly with decreasing temperature below 20 K (cf. Figure 1A). With τe held at 0.211 ps (the value at 80 K when T01 is 0.1 ps), the electron-transfer rate decreases at low temperatures, but the decrease is only about half that seen when T01 is held constant at 0.1 ps (cf. Figure 1B). The residual decrease in the rate at low temperatures when τe is fixed at 0.211 ps (i.e., when τe is short relative to electron transfer but is independent of temperature) can be assigned to the loss of electron transfer from vibrational states with energies exceeding kBT. The dependence of the vibrational thermalization time on temperature has not been measured directly in reaction centers. However, the time constants for damping of the oscillations in the stimulated emission spectrum of P*, which probably reflect dephasing of a vibrational wavepacket on the potential surface of the excited state, are almost independent of temperature between 295 and 10 K.65 The situation modeled in Figure 4A, where T01 increases with decreasing temperature and τe remains more or less constant, thus probably is more realistic than that modeled in Figure 1, where τe decreases with decreasing temperature. One would expect T01 to increase with decreasing temperature because vibrational transitions are driven by oscillating electric fields from the surroundings, which should decrease in amplitude with decreasing temperature. The temperature dependence of the electron-transfer rate constant

Parson and Warshel

Figure 3. Oscillations of the energy gap between P+ BA and P* following coherent excitation of P from its lowest vibrational level at 80 K (dotted curves) or 5 K (solid curves), with T01 ) 1 ps at both temperatures (A) or with T01 ) 1 ps at 80 K and 2.11 ps at 5 K (B). The mean value of the energy gap (∆E° + λ) was subtracted in each case. An initial, very fast decay component that is independent of temperature and is unrelated to vibrational thermalization or dephasing76 is off scale at the top of each plot. Changing T01 as in B keeps the thermalization time τe constant at 2.11 ps.

Figure 4. Temperature dependence of the effective rate constant for electron transfer from P* to BA when T01 is adjusted with temperature to keep τe constant at either 2.11 ps (filled circles) or 0.211 ps (empty circles). ∆E° ) -300 cm-1. In A, P* was prepared coherently by excitation of P from its lowest vibrational level; in B, P* was prepared incoherently with its vibrational levels in a Boltzmann equilibrium at the indicated temperature. The values of T01 used to keep τe ) 2.11 ps are shown with filled circles in Figure 2B. The corresponding values of T01 for τe ) 0.211 ps are smaller by a factor of 10. The situation considered in B is not physically realistic when vibrational thermalization is slower than electron transfer.

calculated under this assumption with τe ≈ 2 ps (Figure 4A) is qualitatively similar to the temperature dependence seen experimentally, although the calculated change in the rate between 80 and 5 K is only about half the observed effect. The rate constant calculated for τe ≈ 0.2 ps decreases substantially below 40 K, in qualitative disagreement with experiment. One way to examine the contribution of hot vibrational states to the dynamics of electron transfer is to calculate the electrontransfer rate for a model in which P* is prepared initially with its vibrational levels in a Boltzmann equilibrium. This is not a

Photosynthetic Electron-Transfer Kinetics

J. Phys. Chem. B, Vol. 108, No. 29, 2004 10479

Figure 5. Temperature dependence of the effective rate constant for electron transfer from P* to BA, after preparation of P* with its vibrational levels populated in a Boltzmann equilibrium at the indicated temperature, for ∆E° ) -100 cm-1 (squares) or -300 cm-1 (circles). T01 was held fixed at 1 ps (A) or 0.1 ps (B). The situation considered in A is not physically realistic.

physically realistic situation if τe is longer than the effective time constant for electron transfer, because vibrational thermalization would lag behind electron transfer. It is, however, straightforward to simulate in the density-matrix model. Figure 4B shows the results of such calculations with τe ) 2.11 or 0.211 ps, and Figure 5 shows similar calculations in which T01 was pegged at either 1 ps (Figure 5A) or 0.1 ps (5B). The difference between the rate constants for T01 ) 1 and 0.1 ps (panels A and B of Figure 5, respectively) stems from the watched-pot effect and differences in the vibrational relaxations of P+ BA , because electron transfer from hot vibrational levels of P* has been removed. Decreasing T01 from 1 to 0.1 ps lowers the electron-transfer rate by about 40% without greatly altering the temperature dependence of the rate although the effect depends somewhat on the energy difference between P* and P+ BA . On the other hand, the difference between the rate constants in panels A and B of Figure 4 can be attributed mainly to electron transfer from hot vibrational states, because here the dynamics of vibronic relaxations are held fixed. (The lack of vibrational coherences in the calculations shown in Figures 4B and 5 results in exponential rather than stepwise kinetics, but coherence in itself has relatively little effect on the overall rate of electron transfer.76,88) A comparison of panels A and B in Figure 4 shows that, in the model considered here, electron transfer from hot vibrational states has a major impact on the temperature dependence of electron transfer. Without this contribution, the rate constant decreases with temperature below 60 K, instead of increasing, even when τe is fixed at 2.11 ps. To this point, we have assumed that P is in its lowest vibrational level before the excitation. Expanding the model to include excitation from a Boltzmann distribution of vibrational levels rearranges the initial populations of the vibrational levels of P* in a temperature-dependent manner, as shown in panels B and C of Figure 6. The temperature dependence of the vibrational thermalization dynamics also changes, as shown in Figure 2. Panel A of Figure 6 shows the resulting temperature dependence of the electron-transfer rate when T01 is varied to keep τe fixed at 2.11 ps. The rate calculated when only the lowest level of P is excited is replotted from Figure 4A for comparison. Including a thermal distribution of vibrational levels of P lowers the rate at high temperatures without affecting the rate at 5 K, making the difference between the rates at 5 and 80 K about twice that obtained when only the lowest level of P is excited.

Figure 6. (A) Temperature dependence of the effective rate constant for electron transfer from P* to BA when P is excited exclusively from its lowest vibrational state (filled circles) or from a Boltzmann distribution of vibrational levels at the indicated temperatures (filled diamonds). T01 was adjusted with temperature to keep τe constant at 2.11 ps in both cases. The values of T01 used are shown with corresponding symbols in Figure 2B. ∆E° ) -300 cm-1. The calculations shown with filled circles included all vibrational levels with energies up to 350 cm-1 above the zero-point levels in P and P* and up to 650 cm-1 in P+ BA ; in those shown with diamonds, the cutoff energies were 375 cm-1 for P and P* and 675 cm-1 for P+ BA . (Raising the cutoff energies by 25 cm-1 decreased the calculated electron-transfer rates by 2% at 120 K and less at lower temperatures.) (B, C) Initial populations of the vibrational levels of P* when P is excited from a Boltzmann distribution of vibrational levels at 5 K (B) or 80 K (C). These plots include vibrational levels of P and P* with energies up to 375 cm-1 above the zero-point levels. The populations of degenerate levels were summed. The dashed lines show the profile of the excitation pulse.

As noted above, the electron-transfer rate given by eqs 1 or 2 increases with decreasing temperature only if (neglecting possible changes in entropy) the energy difference between the reactant and product states (∆E°) is approximately equal to -λ. Figures 1 and 5 include calculations with the density-matrix model for several values of ∆E°, and Figure 7 shows the dependence of the calculated rate constant on ∆E° in more detail. Results are shown for both 5 and 80 K and for excitation of a Boltzmann distribution of vibrational states of P as well as for excitation of P exclusively from its lowest level. T01 was adjusted to keep τe ) 2.11 ps in all the calculations. For comparison, Figure 7 also shows the rate constant calculated with the Marcus equation (eq 1). The density-matrix model gives a broader dependence of the rate on ∆E° than that predicted by eq 1 or 2, and with the parameters used here, the maximum rate does not occur at ∆E° ) -λ, but rather when ∆E° + λ ≈ 250 cm-1. This is because electron transfer occurs largely from vibrational levels above the zero-point level of P*, as we shall discuss below. In all the calculations described above, the electronic coupling factor Vel was taken to be 10 cm-1. The calculated rate constant

10480 J. Phys. Chem. B, Vol. 108, No. 29, 2004

Parson and Warshel makes the calculated rate relatively insensitive to the actual value of Vel, and also somewhat less sensitive to temperature: the ratio of the rates at 5 and 80 K decreases from 1.49 at Vel ) 1.25 cm-1 to 1.21 at Vel ) 30 cm-1 (Figure 8B). This again departs from the behavior predicted by eqs 1 and 2, where changing Vel scales the rate up or down by the same factor at all temperatures. The coupling of the effects of Vel and temperature in the density-matrix treatment probably reflects the inclusion of off-resonance transitions, which increase in importance with increasing Vel and make the rate less sensitive to mismatch between the vibronic levels of P* and P+ BA. Discussion

Figure 7. Dependence of the effective rate constant for electron transfer from P* to BA on ∆E°, the energy difference between the minima of the potential energy surfaces of P* and P+ BA , at 5 K (open circles) and 80 K (filled circles and diamonds). In the calculations shown with filled and open circles, P was assumed to be in its lowest vibrational state before the excitation and the model included all vibrational levels with energies up to 350 cm-1 above the zero-point levels of P and P* and up to 650 cm-1 in P+ BA ; in the calculations shown with diamonds, P was excited from a Boltzmann distribution of vibrational states and the cutoff energies were 375 cm-1 for P and P* and 675 0 cm-1 for P+ BA . T1 was adjusted as in Figure 2B to make τe ) 2.11 ps at both 5 and 80 K. The rate constant predicted by the Marcus equation (eq 1) also is shown for 5 K (dotted curve) and 80 K (dashed curve). ∆E° is the same as ∆G° in the model considered here because the vibrational frequencies are the same in P* and P+ BA.

Figure 8. (A) Dependence of the effective rate constant for electron transfer from P* to BA on the electronic coupling matrix element (Vel), at 5 K (open diamonds) and 80 K (filled diamonds). T01 ) 1 ps at 80 K and 2.11 ps at 5 K (τe ) 2.11 ps at both temperatures), and ∆E° ) -300 cm-1. P was excited from a Boltzmann distribution of vibrational states, and the cutoff energies were 375 cm-1 for P and P* and 675 cm-1 for P+ BA . (B) Ratio of the calculated rate constants for 5 and 80 K.

of 3 × 1011 s-1 at 80 K is somewhat lower than the measured rate constant of 5 × 1011 s-1 at this temperature in Rb. sphaeroides reaction centers. Although the particular vibronic model considered here should be viewed as illustrative rather than as an attempt to fit the measured rate, the calculated rate constants agree well with experiment if Vel is increased to the region of 25 cm-1 (Figure 8A).76 Recent quantum calculations based on crystal structures of the reaction center have given values of Vel ranging from 11 to 37 cm-1.89-91 Adding higherenergy molecular vibrational modes to increase the reorganization energy also would increase the calculated rate. Raising Vel

The density-matrix model appears to capture the main features of the temperature dependence of the initial electron-transfer step in reaction centers of Rb. sphaeroides. If T01 is on the order of 1 ps at 80 K, and is assumed to lengthen with decreasing temperature so as to keep the time constant for vibrational thermalization (τe) in the region of 2 ps, electron transfer speeds up with decreasing temperature. Values of T01 in the region of 1 to 2 ps also give appropriate dynamics for the damping of oscillations of the energy gap between P* and P+ BA at 80 K and lead to stepwise electron-transfer kinetics similar to the kinetics seen experimentally.76 Much shorter vibrational equilibration times appear to be inconsistent with the experimental observations on all these points. Some of the calculations described above assume that P is exclusively in its lowest vibrational level prior to the excitation pulse so that any temperature dependence of the initial conditions is removed from the model. Temperature then enters the model only through its effect on the ratio of the rate constants for downward and upward transitions in which one of the quantized vibrational modes transfers a phonon to or from the surroundings.76 Although these transitions do not change the electronic state of the system directly, the increasing directionality of transfer of phonons to the surroundings at low temperature makes electron transfer effectively irreversible. As shown in Figure 4A, this can cause the net rate of electron transfer to increase with decreasing temperature, even though rapid vibrational thermalization of P* and rapid quenching of the off-diagonal density-matrix elements that connect P* with P+ BA have the opposite effect. However, the calculated difference between the rates at 5 and 80 K is only about half that seen experimentally.1-8 Including a Boltzmann distribution of vibrational levels of P slows the reaction at high temperatures, bringing the temperature dependence of the rate into remarkably good agreement with experiment (Figure 6A). Our explanation for the inverse temperature dependence of the kinetics thus differs from the usual picture, in which the potential energy surfaces of P* and P+ BA cross near the minimum of the P* surface and lowering the temperature increases the probability of finding P* in this region. In the picture suggested here, electron transfer occurs more favorably from higher vibrational levels of P*. Vibrational thermalization of P* is intrinsically unfavorable but occurs slowly relative to electron transfer. Lowering the temperature increases the probability of finding the ground electronic state (P) near its minimum, which shifts the distribution of vibrational levels of P* that are populated by the excitation flash to more favorable energies. In addition, lowering the temperature favors vibrational relaxations that pull P+ BA out of resonance with P*, making charge separation effectively irreversible. Because electron transfer from P* to BA occurs more rapidly than vibrational thermalization, theoretical treatments that invoke

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thermally weighted Franck-Condon factors cannot be used reliably to account for the temperature dependence of the reaction. It is perhaps less obvious that such treatments also cannot be used to determine the reorganization energy. The fact that the optimal value of ∆E° + λ in the density-matrix model is not zero (Figure 7) illustrates this point. The difference between the optimal values of ∆E° + λ predicted by the Marcus equation and the density-matrix treatment can be understood qualitatively on the assumption that the free energy surfaces of the diabatic P* and P+ BA states are parabolic functions with the same curvature along the reaction coordinate. The two surfaces then will intersect at an energy Eq ) (∆E° + λ)2/4λ above the minimum energy of P*. Thus if the excitation flash creates P* with vibrational energy Eex above the minimum energy, and no vibrational relaxation occurs on the time scale of electron transfer, the electron-transfer rate will be maximal when Eex ≈ (∆E° + λ)2/4λ, or

∆E° + λ ≈ 2xλEex

(6)

Equation 6 will tend to overestimate the optimal value of ∆E° + λ because, in addition to neglecting any vibrational equilibration that does occur on the time scale of electron transfer, it neglects nuclear tunneling. In the density-matrix model, the rate of electron transfer from a given vibrational state of P* depends on the vibrational overlap integrals and energy gaps between this state and all the vibrational levels of P+ BA. The reactant and product vibronic substates do not have to be exactly degenerate, although the contribution to the overall rate from a particular pair of states of P* and P+ BA falls off rapidly as the energy gap between these states increases. The overall rate cannot be expressed in a simple analytical form as a function of ∆E°, λ, and Eex, because the number of pertinent states depends on the vibrational structure of the system. It can however, be evaluated numerically for a model with specified vibrational frequencies and displacements, as we have done here. Panels A and B of Figure 9 show how the weighted sum of Franck-Condon factors for electron transfer depends on ∆E° + λ in a simple quantum system with a single, harmonic mode that is excited selectively with either zero or five phonons. The phonon energy (pω) is the same as that used for the lowestfrequency mode in the density-matrix model (30 cm-1). The quantity plotted is

F(m) )

∑n {|〈χn(2)|χm(1)〉|

2

2

exp[-2(n(2),m(1)/w) ]}

(7)

where m denotes the vibrational level of the initial state (0 or 5), 〈χn(2)|χm(1)〉 is the overlap integral of the initial vibrational wave function (χm(1)) with vibrational wave function n of the product electronic state (χn(2)), n(2),m(1) is the energy difference between the same pair of vibronic states (∆E° + (n - m)pω), and w ) 10 cm-1. The factor exp[- 2(n(2),m(1)/w)2] is an arbitrary quasi-delta function that falls below 0.01 when |n(2),m(1)| > 15 cm-1. As ∆E° is varied, vibrational levels of the product state go in and out of resonance with the excited state, causing F(m) to vary with a period of pω. When the lowest vibrational level of the reactant state is excited, the most productive resonances cluster around ∆E° + λ ) 0. Exciting a higher level shifts the productive resonances to higher energies, and the magnitude of the shift increases with the displacement of the vibrational coordinate in the product state (∆). For m ) 5, the optimal value of ∆E° + λ is approximately 214 cm-1 when ∆ ) 2.5 (λ ) 93.8 cm-1) and 295 cm-1 when ∆ ) 3.7

Figure 9. The weighted sum of Franck-Condon factors (F(m), eq 7) for electron transfer coupled to a single vibrational mode with pω ) 30 cm-1 and a dimensionless displacement (origin shift, ∆) of 2.5 (A) or 3.7 (B), when the reactant state is exclusively in either the lowest (m ) 0, dotted curves) or the sixth (m ) 5, solid curves) vibrational level. The abscissa is ∆E° + λ, where ∆E° is the difference between the zero-point energies of the product and reactant states and λ is the reorganization energy, (1/2)pω∆2. The shapes of the resonance peaks depend on the arbitrary function exp[- 2(n(2),m(1)/w)2] and have no particular significance but are sharp enough so that each peak effectively represents a single vibrational level of the product. Panels C and D show the corresponding sums (F(m1,m2)) for a two-mode system with pω1 ) 30 cm-1, ∆1 ) 3.7, pω2 ) 70 cm-1, and ∆2 ) 2.0, when the phonon excitation vector (m1,m2) of the reactant electronic state is (5,0) (C) or (3,1) (D). Note that the abscissa scales in C and D are shifted to higher energy compared to those in A and B.

(λ ) 205.4 cm-1) (see Figure 9). Equation 6 predicts optimum values of 237 and 351 cm-1, respectively. In a multimode system, the density of vibrational levels of both electronic states increases with energy. This tends to smear out the oscillatory patterns seen with a single mode and shift the mean of the distribution to more negative values of ∆E°, as shown in panels C and D of Figure 9. The Gaussian excitation

10482 J. Phys. Chem. B, Vol. 108, No. 29, 2004 spectrum used in the density-matrix calculations causes additional smearing, as does excitation from a distribution of vibrational levels of P. The density-matrix treatment retains the phases of the overlap integrals for various combinations of vibrational levels, which allows for constructive or destructive interferences that are missing in eq 6, but this probably has only minor effects on the calculated rate in most cases.76 The behavior illustrated in Figure 9 can be seen in model calculations by Spears for electron transfer in Co complexes following selective excitation of particular vibrational modes.92,93 However, it is lost, not only in analytical expressions that assume strong coupling and rapid vibrational equilibration in the reactant state, such as eq 2, but also in the quantum mechanical expressions that have been derived by saddle-point treatments for weak coupling (e.g., refs 41, 45, and 94). The wide use of these expressions may have led to the impression that they are more general than they actually are. Most discussions of the energetics of photosynthetic charge separation (see, e.g., refs 4-6, 34, 95, and 96) have been based on the presumption that the rate is maximal when ∆E° ) -λ, which could lead to an underestimate of the value of λ. The measured rate constant in reaction centers from Rb. sphaeroides and related bacterial species increases slightly if ∆E° is made more negative by site-directed mutagenesis of amino acid residues near P or BA, and it decreases if ∆E° is made less negative.4-6,97-105 Assuming that the mutations do not greatly alter Vel or the vibrational modes of the reaction center (which may be incorrect for some of the mutations91,105) these results indicate that the value of ∆E° in wild-type reaction centers is less negative than optimal (see Figure 7). Equations 1 and 2 predict that the rate would decline with decreasing temperature in this situation. Haffa et al.4 have suggested that reactions from unequilibrated vibrational states allow electron transfer to occur rapidly at low temperatures even in mutants in which ∆E° is shifted away from the optimal value, and the results obtained with the density-matrix treatment are in accord with this suggestion. Temperature could affect the electron-transfer kinetics in ways we have not considered, such as by thermal contraction of the protein2,106-109 or by shifting an equilibrium between two discrete conformational states.12,18,110-112 The absorption spectrum of P shifts to the red with decreasing temperature or increasing pressure, possibly because compression strengthens the mixing of charge-transfer states with the exciton states of the bacteriochlorophyll dimer,106-109 and thermal contraction could have a similar effect on the electronic coupling of P and BA. The quantitative importance of thermal contraction is difficult to predict, because the contraction has not been measured directly and probably is anisotropic. However, the calculations shown in Figure 8 indicate that, if the electronic coupling factor Vel is on the order of 20 cm-1 at room temperature as recent quantum mechanical calculations suggest,89-91 the electron-transfer rate would not be highly sensitive to increases in the coupling. The insensitivity of the electron-transfer rate to changes in Vel in the region of 20 cm-1 (Figure 8) has an additional implication concerning the specificity of electron transfer from P* to BA and HA in preference to the pigments on the opposite side of the reaction center’s axis of pseudosymmetry (BB and HB). Although the electronic coupling of P* to P+ BA appears 90,91 this difference to be stronger than the coupling to P+BB, probably is less important than one would expect on the assumption that the rate is proportional to |Vel|2. A difference

Parson and Warshel + between the energies of P+BA and P BB may be sufficient to account the specificity without a large contribution from Vel.37,113-123

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