13545
J. Phys. Chem. 1995,99, 13545-13554
Coherence, Friction, and Electric Field Effects in Primary Charge Separation of Bacterial Photosynthesis Yurij I. Kharkats? Aleksander M. Kuznetsov? and Jens Ulstrup**$ The A. N. Frumkin Institute of Electrochemistry of the Russian Academy of Sciences, Leninskij Prospect 31, 1I7071 Moscow, Russia, and Chemistry Department A, Building 207, The Technical University of Denmark, 2800 Lyngby, Denmark Received: April 14, 1995@
New electron transfer patterns have been revealed by the dynamics of the special pair chlorophyll, the auxiliary chlorophyll, and pheophytin in primary bacterial photosynthetic charge separation. We address here several of these charge transfer features for which new elements of electron transfer theory, disregarded in common theoretical approaches to photosynthetic electron transfer, are needed. These features are, first, an auxiliary chlorophyll state weakly off-resonance relative to the special pair and pheophytin states. Common superexchange views fail in this case, and incorporation of continuous vibrational manifolds in all three electronic states is essential. Secondly, external electric fields may induce transitions between the superexchange and sequential electron transfer modes where the vibrational manifolds ensure that mild resonances rather than divergence appear in the rate constant at resonance. The sequential mode finally incorporates a variety of frictional patterns ranging from vibrationally almost relaxed transitions to strongly dynamically coupled, coherent transitions.
1. Introduction Views on the “ultrafast” primary charge separation in the bacterial photosynthetic reaction center have recently been in a state of notable m0bi1ity.I-l~ This process involves electron transfer (ET) from excited bacteriochlorophyll dimer, P*, to bacteriopheophytin in the L-branch of the reaction center, HL, assisted by the bacteriochlorophyll monomer, BL, spatially located between P and H L . ’ - ~Most data relating to kinetics, magnetic singlet-triplet hyperfine splitting, and P* fluorescence anisotropy for the ET sequence
were previously taken to support single-step superexchange ET. This mechanism implies that the intermediate P+.BpHL state energy is too high and that the state is populated but electronically off-resonance coupled to P and HL. New kinetic and magnetic data have led to revision of this view. Multiphasic kinetics in the picosecond and subpicosecond time ranges for Rhodobacter ~phaeroides~.~.’ l.I3-l5 have been represented by two time constants, viz., 3.5 and 0.9 ps, which drop to 1.4 and 0.3 ps, respectively, at 25 K.I4,l5 Inhomogeneous broadening,’ excited P*-relaxation,’O.’1,13,17-19 and sequential activationless ET with intermediate P+*B;*HL population are all mechanisms formally in line with such a pattern. Other appropriate references to this point are refs 20 and 21. Intermediate state population has also received support from MARY and RYDMR data. These have provided a larger upper limit than before for the P**BL P+-B, energy gap and more suitable ET reorganization Gibbs free energie~.~-~-’ It has also been argued that anisotropic field effects on the P*-fluorescence yields concord better with sequential than superexchange ET when details of the electronic donor and acceptor charge distribution are
-
* To whom all correspondence should be addressed. ‘The A. N. Frumkin Institute of Electrochemistry of the Russian Academy of Sciences. ’ The Technical University of Denmark. Abstract published in Aduance ACS Abstracts, August 1, 1995. @
0022-365419512099-13545$09.00/0
c o n ~ i d e r e d . ~The . ~ intermediate state energy has finally been approached by molecular dynamics s i m u l a t i ~ n , but ~ ~ . the ~~ outcome of such calculations depends rather sensitively on the models used and is not distinctive enough as to the precise BLfunction. BL-population has received recent support from reaction center dynamics, where B L - ~or ~H~-populations~~-~’ were replaced by analogues with longer lifetimes and better spectral resolution. An “ultrashort” BL-population lifetime would reveal new perspectives for the primary charge separation dynamics. First, the extremely fast time evolution implies that the P+.B,*HL state may not evolve to full vibrational relaxation before proceeding to the P+-BL*HL~ t a t e . ’ ~The . ’ ~ two electronic transitions involved, Le. P**B Pf.B- and B-.H BH-, then proceed “coherently”,and relaxation features and electronic coupling of both transitions appear in the rate constant. Secondly, coherence could give oscillatory time evolution in all three states9,28.29 if the intermediate state is weakly damped. Reported oscillatory behavior (0.4-2 ps) of the P*-stimulated emission band in R. sphaeroides R-26 and a R. capsulatus DLL mutant9.30-32and of spontaneous emission in wild-type and mutant R. ~ p h a e r o i d e shave, ~ ~ in fact, been proposed to reflect coherence associated with dephasing and vibrational relaxation. Finally, the P*/HL energy gap of about 0.2 eV and estimated upper limits for the P*/BL gap of 0.07-0.09 e P 7 are low enough that external field induced transition between superexchange and vibrationally coherent ET modes could be envisaged. Electric field effects on the ET steps, eq 1, and on subsequent ET involving the quinone acceptor, QA, are illuminated by several reports. Feher, Okamura, and their associates investigated field effects on the P-band absorption and ET recombination between P+ and Q in membrane i bilayer-fixed centers, exposed to aqueous surrounding^.^^ The rate constant followed a Gaussian dependence over 500 mV, with a reorganization Gibbs free energy of 0.64 eV. Dutton et al. used oriented samples in Langmuir-Blodgett films. They obtained multiexponential kinetics and exponential rate constant variation with the potential This configuration,however, involves solid
-
0 1995 American Chemical Society
-
13546 J. Phys. Chem., Vol. 99, No. 36, 1995
electrodes in contact with stacked films out of the aqueous environment. Boxer et al. studied electric field effects on special pair absorption and emission and on the kinetics of the primary charge ~ e p a r a t i o n . ~ ~Fluorescence -~~ quantum yields and transient HL-absorption show that the rate constant for Hi-formation in eq 1 is reduced by fields up to lo6 V cm-I, but the effects are much smaller than expected from simple models. This also applies to P + Q recombination. The field dependence therefore points to more detailed vibrational frequency dispersion than in single- and two-mode models and to more precise internal field corrections. Field effects on picosecond fluorescence and primary ET rate constants in photosystem IP9 and photosystem I luminescence in spinach chloroplasts40have also been observed. Most theoretical approaches to the dynamics of eq 1 have rested on multiphonon ET theory, applied to a single kinetic superexchange event or two consecutive, equilibrated singleET steps.1-s.10.18.41-46 Field effects are reflected in the activation Gibbs free energy and in the electronic energy denominator of the superexchange ET rate constant. Common superexchange rate constant forms, however, diverge as the three levels approach resonance. In addition, environmental friction in the intermediate vibrationally unrelaxed state is not covered by ET theory in the form commonly used. A more general frame which incorporates both resonance and coherence is therefore needed. We have previously attended to these features in a general Transparent rate constant forms have been obtained by incorporating both high- and low-energy intermediate states in second-order quantum mechanical perturbation theory combined with quasiclassical nuclear wave function^.^^.^^ This formalism incorporates intermediatestate continuous distribution of vibrational levels, which also ensures convergency of the rate constants. The approach can be extended to multidimensional potential adiabatic transition^,^^.^' highfrequency nuclear modes,52and continuous electronic spectra as in adsorbate-assistedelectrochemicalET53or scanning tunnel microscopy of large molecule^.^^ In all these cases classical trajectories on the potential surfaces reflect the coherence of the transitions. In a quantum mechanical view intermediate state wave packets represent the coherence. The formalism has recently been combined with environmental friction. 12,28 This view leads to a more detailed description of the electronic factor and to damped oscillations in all three states. The divergency problem at intermediate state resonance with the donor and acceptor states was approached by vibrational density matrix theory in the three states, in more rigorous quantum mechanical appro ache^.^^,^^ The transition probability could be recast in a form with close relations to Raman and resonance Raman ~ c a t t e r i n g . ~Vibrational ~-~~ dispersion ensures that rate constants are finite at resonance, with no need for ad hoc damping features. In other recent reports Lin reconsidered three-level ET in a one-dimensional system.59 Due to the one-dimensional nature of the model, the rate constant, however, remained resonantly divergent. Marcus and Almeidam investigated the time evolution in coherent two-step ET to a coupled bridge/acceptor unit, using numerical integration of the time dependent nuclear Schrodinger equation. They attended specifically to intermediate state population which can decay faster than in the fully adiabatic limit when the nuclear velocity is large. Hu and Mukamel also investigated damping and resonance in threelevel optical and thermal electronic transitions using the Liouville pathway formalism.6’ Other views of coherent ET are given in refs 62 and 63.
Kharkats et al. In the present report we use the integrated three-level ET formalism including vibrational density matrix theory and coherent reactive attempts with kinetic time evolution in eq 1. In section 2 we overview second-order quantum mechanical perturbation theory in three-level electronic-vibrational systems. In section 3 we add vibrational damping to the intermediate state. Coherence and oscillatory behavior emerge in this limit. Section 4 addresses the primary photosynthetic ET sequence, and we analyze extemal field effects, resonance and offresonance rate constants, and critical voltages where transitions between superexchange and “sequential” coherent ET modes might occur.
2. Electronic-Vibrational Coupling and Resonance in Three-Center Electron Transfer The diabatic ET rate constant in a two-leuel system involving strong electronic-vibrational coupling to an environmental continuum is, by the Fermi golden rule, and the Condon approximation,64
where VHP is the electron exchange factor for direct coupling of the electronic states P* and H, and S(H~)(PI) is the nuclear Franck-Condon overlap factor for the vibrational states 1 and n with the energies EHI and E H ~in the electronic states P* and P+H-, respectively. AVPIindicates statistical averaging with respect to the 1-states; and C, summation, with respect to the n-states. 2nh is Planck’s constant. Equation 2 is, after recasting the &function in integral form,
-
PI
where A G p is the energy gap; Zp, the canonical vibrational statistical sum in the P*-state; k ~ Boltzmann’s , constant; and T, the temperature. This equation is the form from which rate constants can be derived when the nuclear potentials are specified. Another two-level rate constant form useful in the following is65
where @p( 1 - 0) and @H(@are the vibrational density matrices in the P*H- and P+H--state, at the temperatures T/(1 - 0) and T/0, respectively, Hw and HHO,the vibrational Hamiltonians; and 3,the vibrational Gibbs free energy in the P*-state. The rate constant for a three-level system involving the intermediate BL-state in addition to the donor (P*) and acceptor
Primary Charge Separation of Bacterial Photosynthesis
J. Phys. Chem., Vol. 99, No. 36, 1995 13547
physical pattem of the process and the character of the intermediate state population. In particular, the apparent divergency as the energy denominator vanishes is lifted by the vibrational continuum, similarly to the d-function divergency in two-level reactions. The continuum is thus essential for coherent and resonance three-level ET. In return ad hoc introduction of intermediate state lifetime parameters is no longer needed. (4)In the limit of low-energy intermediate states eq 5 applies to a single frictionless intermediate state passage. We return to this issue in section 3. (5) As is common in diabatic ET theory, the initial, intermediate, and final states coupled by eq 5 are the separate electron states. These are P**BL’HL,P+-B?HL, and P+-BL.H;. These states are nonorthogonal, and the time evolution refers to the time evolution in the observables of these three states. Other separation schemes, say adiabatic basis sets, are feasible but less convenient. Detailed handling of eq 5 is reported e l ~ e w h e r e , ~ ~but - ~ ’we summarize some results needed to analyze photosynthetic phenomena. The results rest on the vibrational density matrix formalism, for harmonic displaced surfaces, of the form
AU
tu
Figure 1. Three-level, one-dimensional potential surfaces representative of different processes: (a) high-energy intermediate state representative of superexchange chemical or biological ET; (b) low-energy intermediate state, representative of two-step, coherent chemical or biological ET such as eq 1; (c) ET where the initial nuclear configuration is recovered subsequent to ET, representative of scanning tunnel ET between a set of substrate (I/sUbsu) and tip (Ut,p)potential surfaces through a low-energy molecular adsorbate ( (d) optical threelevel process. This is representative of resonance Raman scattering where the q-mode is a low-frequency solvent mode, the intermediate state is an excited molecular state, and the initial and final states are ground and excited vibrational levels of a Raman active mode in the electronic ground
+
+
U , = 1 / 2 h ~ ( q qo)2 A G , ( 6 ) where o is the vibrational frequency; qo, the equilibrium displacement; and A@, and Ahuo,,, the energy gaps. Potential surfaces of more general form can be handled by quasiclassical nuclear wave which do not, however, incorporate the resonance effects. Recasting the energy denominators in eq 5 in integral form we obtain a triply integral rate constant form analogous to eq
4: (HL) states, K$$, rests on second-order quantum mechanical perturbation theory. The form analogous to eq 2 is47555
Using the explicit form of the vibrational density matrices, Wt: becomes
Y --+
(5)
This equation couples the PI and Hn manifolds via the intermediate electronic state, the vibrational states of which are characterized by the index m. VBP and VHB are the electron exchange factors. Equation 5 is an attractive representation of the three-center sequence in eq 1. The following observations are appropriate: (1) The equation involves the three sets of electronicvibrational states. Like eqs 2 and 3 it is general and applies to arbitrary nuclear potentials (Figure 1). (2) Equation 5 is related to the superexchange form often used in long-range ET in metalloprotein66or binuclear intramolecular ET system^.^',^^ However, the commonly used much simpler form only applies when the BL-surface is well above the donor/acceptor (P/HL) crossing. In contrast, eq 5 applies both to this high-energy limit and to low-energy populated intermediate states, including states in resonance with the donor and acceptor states. (3) The intermediate vibrational states are of little importance in the high-energy superexchange limit far above resonance. As the BL-state approaches the P/HL crossing, the intermediate state vibrational manifold, however, crucially determines the
where fBp
fHB
+ l/,(q;/sinh a ) x cosh a ( l - 26 + 2t, + t2)- P A G p ( 0 - z, - t2) = -1/2q: coth a + ‘/,(q:(sinh a) x
= -‘/,q:
coth a
cosh a(1- 28) - B A G , (9) fHBp
= (2q: sinh a cosh a (z, - t2>x sinh a ( l - 0
+ t,+ t 2 )sinh a0
and a = 1’2Pho.Equation 9 is still formally divergent due to the , B A ~ , ( z l- z2) term unless phenomenological “dephasing parameters” are added. The divergency can, however, be lifted by vibrational “coarse graining”55or by density matrix expansion to include quantum mechanical correction^.^^ Either step
Kharkats et al.
13548 J. Phys. Chem., Vol. 99, No. 36, 1995
crossing counted from the P* equilibrium energy, and AU* is the energy gap to the intermediate state (Figure 2 ) . When m < A a n, the intermediate state is populated and the rate constant is
+
+
(n - m Aa)’ 2A
t” +
I
e e e,
U A = v“,‘ = (E, Au0,,)2/4Es when U y > correspond= A G P (E, ing to the P*BL crossing, while UA = AU&)*/4E, for v“,‘ < corresponding to the second crossing. Equation 13 includes two electronic factors and two LandauZener velocities (= (ksr)’/* and = (IUBP - UHB~)”*).This reflects the single-attempt and vibrationally coherent character of the process, clearly distinct from sequential single-ET steps. This form is therefore appropriate to the “ultrafast” processes in eq 1. A final analytical form emerges when the two crossings of a populated intermediate state are in resonance, Le. v“,‘ = (Figure 2b).
W
tu
+
+
e
I
C
Figure 2. Three-level configurations represented by eq 10: (a) offresonance configurations; (b) two levels in resonance (eq 15); (c) all three levels in resonance.
reflects vibrational dispersion. By the former procedure the rate constant can be reduced to the singly integral form
@$
= @ 3 d & ) ( v ~ ~ ) 2 ( v ~epx)p2( - a A & J h o ) F
This form exhibits a “mild” resonance where the logarithmic argument is on the order a-2% ( k ~ T / h >)>~1. Equations 12- 14 represent field and other Gibbs free energy relations in broad energy gap ranges, while the general eq 10 must be used close to resonance. Equations 13 and 14 can be combined with vibrational damping, to which we now attend.
9& = ( l / a ) i ( - l ) m + ’ x d t exp(-2A cosh a sin2 r) x sin2 ’/$ (sin ‘/2t)2pcos(mt
+ A sinh a sin t )
where
n = IA&,l/ho
and m = (A$,
- A&,)/b
(11)
are regarded as integers. A = ‘/2q2dsinha represents the reorganization energy for ET through the intermediate state, i.e. E, = “2wq20 = A d t o . Equation 10 can be reduced to analytical forms in broad offresonance regions. Common superexchange corresponds to the condition m A a n and gives
+
exp(-”’PAL$,
where UHP= (4E,
- 2Aa’
+ A$p)2/16E,
- n2/2A)=
is the energy at the P/HL
3. Frictional Effects and Multiple Reactive Attempts at the BL/HL Crossing The coherent ET formalism extends to general potential surface configurations in off-resonance regions, to dissipative effects, and to the adiabatic limit. Dissipation in the overdamped limit finally provides the condition for a single coherent transition in the intermediate state, corresponding to the purely quantum mechanical result in section 2. Dissipative effects lead to new features of importance for eq 1. One is less conspicuous diabatic behavior of the B ~ H L BL*H, transition due to the HL transitions. A second feature is that manifold of BL damped oscillatory time evolution emerges when the energy is weakly dissipated. The three-center rate constant from the equilibrated initial state, via a populated intermediate state including the full manifold of transitions, has the form:’2,28
-
-
Primary Charge Separation of Bacterial Photosynthesis
J. Phys. Chem., Vol. 99, No. 36, 1995 13549
I"
the Stokes shift for special pair excitation and A@,, the energy gap between BL and the ground state of P if motion starts at the configuration created by photoexcitationby exciting light of frequency v. Equation 20 gives
n(E) = (O&nr) ln(Eo/AGB) - 1/2
(21)
-
Figure 3. System trajectory for passage from the P- to the HL-state. PHBincludes the complete manifolds of attempts above the appropriate
where A G , is the barrier height for the BL HL step. We now apply eqs 16-22 to the potential surface configuraP* tions representative of eq 1. By omitting the reverse B; reaction, eq 16 takes the form.
crossings.
= PBp[ 1 - (1 - pHB)n(n+l]
,&p
energy; a n d h p , the overall transition probability (dimensionless) incorporating all individual transitions at the two crossings. The integration limits depend on the frictional behavior of the Bi-state and the potential surface configuration. Equation 16 thus reduces to the formalism in the previous section for a single BJHL crossing and friction-free motion. h p has the form n(E)
/BH~= p B p ~ H B z ( 1- ~ B p ) ~ ( lPHJk=O
(16)
-
where n(E) is the number of attempts at the BC HL crossing. All energies are counted from the bottom of the BL-well. h p incorporates the whole range from adiabatic (PBP, PHBx 1) to strongly diabatic (PBP, PHB
