J . Phys. Chem. 1988, 92, 26962701
2696
phase, where the predominant interaction is donation from ethylene's filled *orbital into the empty 2gorbital of the carbene; and, at a closer approach, a nucleophilic phase, where the carbene's sp2 hybridized lone pair tilts down to interact with the remains of the r*orbital of ethylene, bringing the carbene hydrogens up to their final positions in the cyclopropane ring. However, it is clear that even early on in the electrophilic stage, something beyond 2p donation is controlling the rotational preference simple n of the incoming 'CH,. Without further analysis, the logical assumption is that the nucleophilic interaction is significant even at long 1CH2olefin distances.

Conclusions
The reaction of triplet methylene with acetylene to give triplet vinylcarbene is calculated to proceed with an activation energy of 14 kcal/mol by ab initio methods, or 11 kcal/mol by MNDO. The singlet reaction proceeds directly to cyclopropene with no barrier on the ab initio potential energy surface. MNDO, in contrast, calculates a barrier of 7 kcal/mol for the singlet reaction but again leads directly to cyclopropene. The two methods agree reasonably well on the geometries and force constants in the triplet transition structures 3 and 4. In particular, all methods agree that, as long as the sp2 orbital of an approaching 3CH2is directed toward one of the acetylene carbon atoms, there is little or no preference for 3CHz rotational orientation. Thus, there is no significant CcCi double bonding in the transition structure leading to triplet vinylcarbene 5. Furthermore, as estimated by the increase in rotational preference along the collapse path, the development of nbonding is slow around the transitionstate distance. This doublebondforming addition reaction, while obviously concerted, is not synchronous. The triplet addition barrier seems high compared to those for methyl radical additions, and in light of the known rapidity of 3CH2quenching by acetylene. How large could our errors be? Available rate data'618~28 and our preliminary resultsi2 at the MP2/321G level suggest that the 2CH, HCCH and 2CH3 H,CCHz additions are energetically similar to the present system. Yet the experimental activation energy for this "simple" radical addition is firmly established at ca. 7.7 kcal/mol. It seems unlikely that further basis set improvement or CI corrections based on our singleconfiguration U H F wave function could halve the 13.9 kcal/mol barrier we predict. For our U H F wave function, the value of S2 in the higher level calculations never exceeded 2.2. This indicates relatively little contamination of the triplet state and therefore minor energetic errors from this source. However,
+
+
Sosa and Schlegel have recently reported that annihilation of spin contamination lowers their calcuiated activation barrier for attack of HO' on acetylene by a whopping 9.6 kcal/m01.~~Possible corrections for thermal effects were briefly explored, but if anything, their inclusion would raise the computed activation barrier. In an analogous study of the reaction between a triplet oxygen atom and acetylene, Dunning et al. report4' an activation barrier of 10.5 kcal/mol, which exceeds the experimental 3 kcal/mol value substantially. Having explored the effects of various levels of CI treatment, starting from their singleconfiguration R H F wave function, these authors conclude that multireference configuration S C F calculations with C I (MCSCFCI) may be necessary to describe this deceptively simple reaction properly. Similar considerations apply to the reaction of triplet methylene and acetylene. Alternatively, the mechanistic assumptions and methods employed in the experimental study of such gasphase radical reactions may require reexamination. A further theoretical study comparing the behavior of T H 2 and 2CH3in reaction with ethylene and acetylene is currently underway. We look forward to continuing experimental and theoretical developments in this field over the next few years.
Acknowledgment. We gratefully acknowledge support by the National Science Foundation via grant C H E 83 18345 awarded to Professor Maitland Jones, Jr. and by Princeton University via the Charlotte Elizabeth Procter Honorific Fellowship awarded to J.E.J. We warmly thank M. Jones, Jr. and Evelyn Parker for patience, valuable comments, and assistance with this manuscript. Thanks are also due to R. P. L'Esperance, D. B. Kitchen, R. A. Pascal, M. S. Platz, and I. Shavitt for fruitful discussions. Registry No. 1, 2465567; 2, 74862; 5, 19527083; 12, 2781853. (40) Sosa, C.; Schlegel, H. B. J . Am. Chem. SOC.1987, 109, 4193. (41) Dunning, T. H. Jr.; Harding, L. B.; Wagner, A. F.; Schatz, G. C . ; Bowman, J. M. In Comparison of Ab Initio Quantum Chemistry with Experimenrfor Small Molecules; Bartlett, R. J., Ed.; D. Reidel: Hingham, MA, 1985; p 67. (42) Note Added in Proof. An optimization of 4 at the MP3/321G level has further validated the approach outlined here. Negligible geometrical differences between MP3/321G and MP2/321G structures were found, except for r(C,C,) which lengthened by 0.038 A (from 2.1 10 A to 2.148 A), similar to the 0.030A increase in the 'MP3/631G*" case. Further basis set improvement also has essentially no effect: MP4(SDTQ)/631 lG**// 'MP3/631G*" calculations give E, values of 14.6 kcal/mol, only 0.10.2 kcal/mol lower than with the 631G' basis; and optimization of 4 at the UHF/6311G** level yielded a geometry almost identical with that at UHF/63 1 G*.
ElectronTransfer Pathways in the Primary Event of Bacterial Photosynthesis Arieh Warshel,*+ Steven Creighton,' and William W. Parson* Department of Chemistry, University of Southern California, Los Angeles, California 90007, and Department of Biochemistry, University of Washington, Seattle, Washington 981 95 (Received: July 9, 1987) The photochemical reaction in bacterial photosynthesis is the transfer of an electron from a bacteriochlorophyll dimer (P) to a bacteriopheophytin (H), possibly by way of another bacteriochlorophyll (B). We explore the mechanism of this reaction by calculating quantummechanical matrix elements for the electronic coupling of the exciton states and intermolecular chargetransfer (CT) states of the photosynthetic reaction center. The calculations are based on the crystal structure of the Rhodopseudomonas uiridis reaction center and on semiempirical atomic resonance integrals. Five possible mechanisms for charge separation are considered. The matrix elements for a pathway via the intermediate CT state P'B are consistent with the experimentally measured kinetics of the reaction, but this pathway requires a remarkable stabilization of the CT state by the protein. An alternative pathway via B'H could be more favorable energetically, but has somewhat smaller coupling elements. The initial electrontransfer steps of bacterial photosynthesis occur in proteinpigment complexes called "reaction centers" (see University of Southern California. *University of Washington.
00223654/88/20922696$0l.50/0
ref 1 for a review). The molecular structures of the reaction centers of Rhodopseudomonas viridis and Rhodobacter (1) Parson, W . In Phorosynthesis; Amesz, J., Ed.; Elsevier: Amsterdam, Netherlands, 1987; pp 4361.
0 1988 American Chemical Society
ElectronTransfer Pathways in Bacterial Photosynthesis
The Journal of Physical Chemistry. Vol. 92. No. 9, 1988 1691 states (aand 8) can be written as
Here HpBis the Hamiltonian matrix element describing the electronic interaction of the two states; ks is the Boltzmann constant; A is the solvent (protein) reorganization energy; the vector in = (ml, m2, ..., m.) represents the vibrational quantum numbers of all n intramolecular normal modes of the donor and acceptor molecules; . , S is the FranckCondon factor for the transition from vibronic level m in a to in' in (3; and AG'm,m,is an effective activation barrier (see, for example, ref 1115). The activation barrier is Figure 1. Spcci;,l 1p.tir uf R C K s (l', .mtl IPu1. 1%. ; i n t i II t h c ructiiin ceiiter of N j w ririndi.r. Thc sphcrch rcprcrenl the cwrficiems of the atomic p, orbitals far the lowest unoccupied molecular orbital of each chromophore. the radius is proportional to the mdulus of the cwfficient, and the shading denotes the sign. Dotted lines connecting some of the chromophores show the pairs of atoms tha make the largest contributions to the matrix element^ H , * and H2,.
sphoeroides have been solved recently by Xray diffraction:+ making it possible to explore the detailed molecular mechanism of one of the most efficient systems in bioenergetics. Absorption of light by the reaction center is followed by an extremely fast (3.0 i 0.5 ps) separation of electrical charge.7I0 The reaction cnn be written

PBH k P'BH P+BHwhere P designates a special pair of bacteriochlorophyll (BChl) molecules that are located particularly clase together (PI and P,), H is a molecule of bacteriopheophytin (BPh). and B is an "accessory" BChl that sits approximately in between P and H (Figure I). H is located about 16 A from P, (centertocenter) and about I8 A from P,. Although B seems likely to play an important role in the movement of an electron from P to H, recent kinetic give no evidence for the formation of P+Bas a distinct intermediate. If P'B does form in the initial electrontransfer step, an electron must move from B to H t ~ ) rapidly for the intermediate state to be detectable. The reaction center also contains a fourth BChl and a second BPh, which are structurally related to B and H by a 180' rotation about an axis of local symmetry;2d the roles of these additional pigments are unclear. The present work explores the primary chargeseparation process by converting the crystallographic structural information into quantummechanical coupling terms, which depend strongly on the overlap of the rorbitals of P, B and H . The calculated coupling terms support an involvement of P'Bas an intermediate but raise new questions about the control of the energies of the chargetransfer (CT) states by the protein environment, Theoretical Approach The rate constant for an electrontransfer reaction between two _____~ ~~
(2) Deisenhofcr. J.: Epp. 0.; Miki, K.; Huber, R.; Miehel. H. J. Mol. Bid.
1984.. IRO ~ .7R579R ~ . ~~~~
. ~ ~
(3) Deisenhofcr. J.: Em. 0.:Miki. K.: Huber. R.: Miehel. H. Nature
, , . ~ ~ . ~ ~ .O .. I.Mol. Bioi. 1985. ~
(6) Allcn. J P ; Fchrr. G :Ycain. T. 0.:Re%. D.: Dcwnhaler. H.: Hubcr. R. Prm. l'orl Arad Sri U.S A 1986. 83, 85R98593. 17) Hollm. D , Hop"n. D: Windor. hf W :Schcnek. C. C.. Parson. W W :\ f ~ p A% , Fork. R L ; Shmk. C V . Birrhim Bwphys. A m 1980. ' 0 , 4*1.117 _,,. (8) Wwdbury. N. W.: Beckcr. M.:Middendarf, D.: Parson, W. W. Bio
chemistry 1985. 24, 75167521. ( 9 ) Martin. I.L.: Breton, J.: Hoff,A.: Migus. A,; Antonelti, A. Proc. Nml Acod. Sci. U.S.A. 1986.83, 957961. (10) Breton. J.: Martin. J.L: Migus, A,; Antonelti. A,: Orsag. A. Pmc. Natl. Acad.Sci. U.S.A. 1986.83, 51215125.
AG*m,m,= [AGO  Eku,(m:  m,)
+ AI2/4A
(2)
where AGO is the standard free energy difference between a and 8, and the sum is taken over all of the vibrational normal modes. An evaluation of k, from actual structural information requires a direct calculation of Hm6and AG', which is more difficult than simply fitting the parameters in a phenomenological rate equation to the observed kinetics. However, the primary electrontransfer steps in the reaction center are activationless processes; at 71 K, electron transfer from P to H appears to be even slightly faster for the most effective than it is a t 295 K.8 This means that AG'., vibronic channel is much smaller than k,T. Because S , for this channel probably is close to 1." ea I can be simdified to the approximate expression ken = (H,s/h)2(Th2/ksTA)'/z = 1.65 X 10't(H.6/10)2(200/A)'~2
s'I
(3)
a t 295 K, with Henand A in units of cm'. The protein reorganization energy A is not known, but it is likely to be between 200 and 2000 cm'. When additional information becomes available on the crystallographic coordinates of the protein, it should be possible to calculate both A and the intramolecular Franckxondon factors directly from the s t r u c t ~ r e . ' ~ ' ~ In ' ~ the present paper we focus on the evaluation of Hen, which should depend primarily on the coordinates and molecular orbitals of the chromophores. We shall use eq 3 only as a guide to whether the calculated matrix elements are of the correct order of magnitude lo account for the measured rate of electron transfer. In order to translate geometrical information about the chromophores into interaction matrix elements, one has to choose a convenient, yet physically meaningful representation for the states 01 and p. Although this selection of the zeroorder states is arbitrary, it is helpful to use a representation that gives relatively small effective coupling terms, which are easiest lo integrate in solving the time dependence of the system. Note in this respect that one could start with an adiabafic basis set (a set of states that diagonalize the complete interaction Hamiltonian of the system) and then evaluate the time dependence of the system by considering the nonadiabatic coupling termst' of the form (J.elWJar). It is, however, more convenient to choose zeroorder diabatic states, which do not diagonalize the Hamiltonian but which correspond to clear physical intuition, and lo switch to the (11) K u b , R.: Toyozawa. T. Pmg. Theor. Phys. 1955, 13. 16+182. (12) Washel. A. Prm. Notl. Acod. Sci. U.S.A. 1980. 77, 31053109.
(13) Marcus. R.: Sutin. N. Biochim. Biophys. Acto 1985.811, 265322. (14) Warshcl. A,: Hwang. J.K. J . Chem. Phys. 1986. 84. 49384951. ( I S ) Hwsng. J.K.; Warrhc1.A. J . Am. Chem.Soe. 1987,109,115720. (16) Warshel, A. Notum (London) 1976,260,679683. (11) Churg. T.: Weisr. R.; Warshcl, A.:Takano, T. J . Phys. Chem. 1983. 87. 16831694. (18) Warshel.A.:Schlmscr. D. A.Proc. Nml. A e d S c i . U S A . 1981. 78. SS64SS6U. ... (19) Churg, T.: Warshel. A. Biochemistry 1985, 25. 16151681. (20) Wmhcl. A. In Elcctron Trompon end Oxygen Utilirotion: Ho, C.. Ed.: Elsevier: New York. 1982: pp 111115. ~~~~
2698
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988

TABLE I: Contributions to the ElectronTransfer Matrix Element for P* P'B0.7 13
Warshel et al. chromophores also contribute to P* but do not enter into the matrix element that mixes P* with P'B. The values of B", depend to some extent on assumptions concerning the energies of the CT transitions of P, but the calculated values of the overall matrix elements for electron transfer are relatively insensitive to these
0.696
assumption^.^^ 0.650 0.009 0.005 0.035
6.736 4.388 9.438 3.007


0.127 0.0 14 0.030 0.018 0.245 0.001 0.032 0.023 0.381 0.049 0.043 0.023
13.360 2.203 1.024 5.963 0.489 0.000 2.296 0.000
W


7.377 0.000 10.408 0.000

6.297
2.688 5.89
"Entries shown as  are identically zero; those shown as 0.000 are small but nonzero. adiabatic representation only if the coupling of the diabatic states is too large. To define a zeroorder diabatic basis set, let 9,designate an excited state (P*) that includes excitonic contributions from the local XT* transitions of all six chromophores, along with the C T transitions in which an electron moves only between the two BChl's of P. Let \k, refer to a diabatic C T state that involves reduction of B or H. We shall designate three particular C T states of this type more specifically as \k2 = P'B, \k2, = B"H, and \k3 = P'H. Our initial state now can be written asZ1 \k, = CBai\ki i
(6)
W
where the coefficients B@,are obtained by solving the interaction submatrix2I for the specified C T transitions. We take our second diabatic state, \k2 or P'B, to be the lowest energy combination of transitions in which an electron moves from PL or PMto B. Its dominant components35involve electron transfer from the highest filled molecular orbital of PL or PMto the first empty orbital of B. The coefficients BP, for these two transitions are listed in Table I under CT(PLB)l and CT(P,+B)l. \k2 also includes six additional C T transitions, which involve removal of an electron from the second highest filled orbital of PL or PMor transfer to the second lowest empty orbital of B. These higher energy C T transitions are included in the calculations described below, but their coefficients are relatively small and are not shown in Table I. The additional diabatic C T states \k2, and 'k3 are defined similarly. Thus \k3(P'H) is made up of a linear combination of the eight C T transitions in which an electron moves from PL or P M to H. The Hamiltonian matrix elements that mix \k, and \k, are given by''
1.032 0.621 0.289 0.461

= CBBwQw= CBBw4Jwlw2
*p
0.446 0.968 2.082 0.199

0.571 0.053 0.019 0.037
The diabatic states that involve reduction of B or H can be described as

(4)
where the coefficients B*i are obtained by diagonalizing the interaction Hamiltonian P and the local XA* excited states of the other BChl's and BPh's, and the \k, are given by
HmB=
C Bu,BswU,,w= 1.w
(21) Warshel, A,; Parson, W. W. J. Am. Chem. SOC. 1987, 109, 614361 5 2 . (22) Parson, W. W.; Warshel, A. J. Am. Chem. SOC. 1987, 109, 6 1526 163.
BaIBowC,,NA,,w
(7)
where U,+ = ($,ldQw). The A , , are matrix elements that connect the individual CT transitions that make up \k, with the individual local X  X * transitions or the individual C T transitions that conExpressions for A,,whave been developed in ref 21: tribute to
*,.
Ai,w =
Z ( ~ n l , w l ~ n 2 , r ~ w2 ,b sn 2 , w ~ n l , r u w l , s ) P s i
(8)
5,1
where the u's are the atomic expansion coefficients for the donor and acceptor molecular orbitals and P, is the resonance integral between atomic pz orbitals on atoms s and t. With straightforward modifications, eq 7 and 8 also describe the matrix element for mixing two C T states, \k, and \kv. The matrix element for the backreaction from C T state 'k, to the ground state ('k.,) is given by
HBy =
C~~w~~1'2C~w2,sL'w1,tPrll (9) SJ
W
Here &l.n2 is a wave function for singlet excitation from molecular orbital & to &2. For local transitions of the chromophores, the coefficients Ci,Nare obtained by diagonalizing configurationinteraction matrices for the isolated monomers. If one considers the top two filled molecular orbitals and the first two empty orbitals, each molecule has four such local transitions, Qy, Qx, B,, and By. The pair of BChl's that make up P also has eight CT transitions, for which 4Jnl and &2 are on separate molecules. The coefficients Pi have been evaluated recently for the excited states of the Rps. viridis reaction center.21s22 The treatment accounted reasonably well for the measured absorption, linear dichroism, and circular dichroism spectra of the reaction centers and accounted qualitatively for the sensitivity of the longwavelength absorption band to external electrical fields. Some of the coefficients for the lowest excited state (a = 1 ) are listed in Table I. The table includes only the transitions that are important for electron transfer from P to B; local excited states of the other
i,N,w
Reaction Channels ( i ) Hopping of an Electron via B p*
 kll
P+B
kl3
PfH
The most obvious mechanism for charge separation in the reaction center is for an electron to hop from the special pair of BChl's in the excited complex (P*) to B and from there to H (Figure 2). This twostep sequence requires that the intermediate state, P'B, lie below P* in energy. Table I lists the calculated values of U,,+that contribute to the interaction between P* and P'B. The weighted sum over the U,,+(eq 7) gives an overall matrix integrals that underly this element ( H 1 2 )of 5.9 cm'. The coupling term are highly sensitive to the locations of the donor and acceptor molecules, because the resonance integrals Psi fall off exponentially with the interatomic distances.21 If B is displaced by 0.5 A along the axis connecting ring I of this molecule with ring I of P,, the calculated value of IH121increases to 15.6 cml. Displacements of 0.5 8, are within the 3A resolution of the crystal s t r ~ c t u r e and ~  ~might occur as a result of vibrational motion of the protein. In the Rps. viridis reaction center, a conjugated carbon atom of an ethylidine substituent on ring I1 of B is located about 3.8
15
5.9
I
I
:;::m 10500 1040
The Journal of Physical Chemisfry, Vol. 92, No. 9, 1988 2699
ElectronTransfer Pathways in Bacterial Photosynthesis
10 5 1
ENERGY (cm')
2.5
Bt H
v
2.1x1O5
I
I
I
I 51
24
I
*.O
I
I 
Figure 2. Calculated matrix elements (in cmI) connecting diabatic states of the reaction center. The diabatic states are combinations of different exciton states and CT states and are labeled as described in the text.
Superexchange mechanisms are not shown, because the matrix elements for these depend on the energy gaps between the states. The rate constants for electron transfer depend on the squares of the matrix elements. The decay paths to the ground state (P) are slowed by their small FranckCondon factors, which are not represented here. %, from two of the carbon atoms of rings I11 and V of PL (Figure l ) , and the interactions of these atoms make the largest contributions to the overall matrix element. If the ethylidine carbon of B is omitted from the calculations, H12decreases to 0.6 cm'. Note, however, that the resonance integrals in eq 8 are scaled by the corresponding molecular orbital coefficients, so that the major contributions to the matrix element do not necessarily come from the closest pairs of atoms. The matrix element for the second step in the pathway (H23) is calculated to be 15.0 cm'. As for the first step, this result is highly sensitive to the positions and orientations of the electron carriers. Its largest contributions come from the interactions of a carbon atom in ring I of B with the corresponding carbon in ring I of H (Figure 1). Movements of either B or H by 0.5 8, could change the matrix element by a factor of 2 to 3. HI2and H23 are indicated in Figure 2, along with the matrix elements for several alternative pathways that will be considered below. The finding that the matrix element for the second step is greater than that for the first is consistent with the fact that the putative intermediate state P'B cannot be detected experimentally.'1° The intermediate would decay as rapidly as it is formed. The calculated matrix elements also are reasonably close to the values that are required to explain the experimentally measured rate constant of about 3 X 10" sI. Using eq 3 and taking X to be 200 cmI, a value of 5.9 cm' for H12gives k12 0.57 X 10" sI; moving B by 0.5 8, so that H12increases to 15.6 cmI gives k 1 2 4.0 X 10" SKI. The major problem with the mechanism is associated with its energetics: it is hard to see how P+B could lie below P* in energy. P* itself is already a mixture of exciton states and CT states including PM'PL and PMPL', and the absorption spectrum of Rps. viridis reaction centers indicates that the energies of these C T transitions are aboue the energy of the overall state.22 P'B would seem likely to be even higher in energy, because the distance between the charged species in this radical pair is greater than the distance between PM and PL. The two BChl's of P are centered about 7 8, apart, whereas B is centered about 10.5 8,from PL and 11.6 8, from PM. (The edgetoedge distances also are greater in the case of P+B.) In addition, a hydrogen bond from a threonine residue to the ketone group on ring V of P, may stabilize PM+PL,4,22 whereas there is no amino acid in a position to form such a bond to B.4 However, the energetics of P'B needs to be explored in detail by converting the Xray structure of the protein to chargestabilization energies, taking into account the effect of bound water molecules (as, e.g., in ref 19 and 37). (ii) Direct Transferfrom P to H

P*
k13
P'H
The matrix element for direct transfer of an electron from P to
5
10400
100 100
100 100
0
0
0
100
100
AEi2 (cm1)
Figure 3. (A) Adiabatic energies of the combined states obtained by diagonalizing the Hamiltonian of P* and P+B,as a function of the The energy difference between the two diabatic basis states (alz). energy of P* is held constant, and the energy of P'B is increased from left to right. (B) Matrix element coupling P'H with the combined state that consists predominantly of P*. This is \k,+ in the region AE,, C 0, and k\, in the region AEI2> 0.
H can be calculated in the same way, by letting 9,be P* (as in the first step of the indirect pathway just considered) and taking \kg to be P'H (as in the second step). The calculated matrix element (HIJ is 2.2 X cm', which is too small to contribute significantly to charge separation in the reaction center. The calculated rate of direct hopping is about lo8 times below the measured rate of electron transfer. The semiempirical resonance integrals that we have used21are parametrized only for interatomic distances up to about 4 8,and probably underestimate longrange interactions between P and H through the protein or solvent. At large intermolecular distances ( R ) ,experimentally measured matrix elements for electron transfer reactions in glasses vary as exp(aR) with a 0.6 8,1.23If &, is assumed to decay as exp(0.6rS,) for interatomic distances (rSJ greater than 4 A, H13 increases to 0.09 cm', which is still negligible compared to H12and H23. This treatment of PSIchanges the calculated values of HI2,H23, Hlzsand Hy3 by less than f20%, because these matrix elements are dominated by shortrange interactions. ( i i i ) Superexchange

[P*
 P'B1
k123
P'H
In the "superexchange" mechanism, coupling between P* and P'H is evaluated by considering the initial state to be the combined system of P* and P'B. This can be done by diagonalizing the Hamiltonian for all of the transitions that make up P* and P'B (24 local transitions of the 6 chromophores, 8 C T transitions of P, and 8 C T transitions in which an electron moves from P to B). Combining P* and P'B gives two new diabatic states, (Figure 3A). Because the direct interaction of P* with P'His negligible, the matrix element for electron transfer from one of the combined states to H is proportional to the contribution that P'B makes to the combined state, which depends on the energy difference (AE12)between P* and P'B. The coupling of P* to P'H reaches an upper limit at AE,, = 0, when the combined wave functions become = 21/2[9p. f SP+,]. An upper limit for the electrontransfer matrix element in the superexchange mechanism is thus
I 2'I2H23 = 10.6 cmI
(10) where H23 is the matrix element for the interaction of P'B with P'H (the second step of mechanism i). Unlike mechanism i, the superexchange mechanism can operate when the energy of P+B is above that of P*. However, the coefficient of P+B in the combined state that is closest to P* in character drops quickly as AE12 departs from zero. HI23 thus decreases rapidly as lAE121increases (Figure 3B). In the limit of a large energy difference, the coefficient is approximately (HI*/ AEI2),so that Hi23
(23) Miller, J. R. In Antennas and Reaction Centers of Photosynthetic Bacteria, MichelBeyerle,M. E., Ed.: SpringerVerlag: Berlin, West Germany, 1984; pp 234241.
2700
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 H123
%
(1 1 )
(H12/m12)H23
This gives a very small coupling if lAEI21is significantly positive (e.g., for lAE121= 100 cm', IH123( i= 0.8 cmI). Note that the splitting between the energies of the combined states (Figure 3A) reaches a minimum of 2HI2(about 11.8 cmI) when AEI2= 0. Using this adiabatic representation of the combined system provides an alternative way to examine the interaction of the two diabatic states (P* and P'B). The same approach can be used to examine the interactions between P'B and P'H, or the superexchange interactions between the combined system and P'H. A discussion of the matrix elements from this point of view has been presented elsewhere.24 (iu) Initial Transfer from B to H p*
 kiy
B+H
kY3
P+H
Another possible mechanism of charge separation is an initial hopping of an electron from B to H , followed by an electron movement from P to B (or a "hole transfer" from B to P). This pathway depends on the fact that P* contains contributions from the excited states of B and H , in addition to the excited states The excited states of B and H are strongly of PL and PM.22,2425 coupled to the C T state B'H (q2).The overall matrix element for the first step in this pathway (HI2,)is calculated to be 2.5 cm', which is about half that obtained for mechanism i. The limiting factor for the formation of B'H is that the calculated contributions of the excited states of B and H in P* are only in the range of 12%, under a variety of a s s ~ m p t i o n sconcerning ~~ the energies of the basis sets.22 (In Table I, the coefficient for the Qy excited state of B in P* is about 0.13; the relative contribution of the local state is given by the square of the coefficient, or 0.017.) As far as energetics are concerned, this channel has no obvious difficulties. The energy of B'H seems likely to be similar to or below that of P*, because the reduction potential of BPh is substantially less negative than that of BCh1.26 The matrix element calculated for the second step of the route via B+H is 1 1 . 1 cm'. As in mechanism i, this is greater than the matrix element for the first step, so the intermediate state would be difficult to detect experimentally. Considering the sensitivity of the matrix elements to the geometry of the complex, this pathway appears to be a viable alternative to pathway i. In a paper that appeared after ref 25 and after this manuscript had been submitted, Fischer and S ~ h e r e have r ~ ~ emphasized the attractiveness of the holetransfer mechanism. Although their conclusions are generally similar to ours, some of the matrix elements that they present are larger. There are several differences between their analysis and ours. First, Fischer and Scherer consider the coupling between orbitals rather than states, which is equivalent to neglecting the summations over N , i, and w in our eq 7. A consistent treatment of configuration interactions can lead to a significant reduction of the calculated coupling terms. Second, to model the dielectric constant, Fischer and Scherer increase the atomic resonance integrals for distances greater than 1.7 A. This is at odds with the previous parametrization of p,, in the range of 3.5 8, and seems incompatible with the spectroscopic properties of the reaction center.21.22Fischer and Scherer also discuss the overlap of nonconjugated atoms, which are not treated explicitly in our aelectron approach. Superexchange via B+H provides an additional possible mechanism of electron transfer from P to H. This has a maximum coupling constant (Hl23) of about 7.8 cm' (2'i2 X 1 1 . 1 ) if the energies of P* and B'H are equal. The width of the resonance (24) Parson, W. W.; Creighton, S.; Warshel, A. In Primary Reactions in
Photobiology; Kobayashi, T.,Ed.; SpringerVerlag: Berlin, West Germany, 1987; 4351. (25) Parson, W. W.; Scherz, A.; Warshel, A. In Antennas and Reaction
Centers of Photosynthetic Bacteria; MichelBeyerle, M. E., Ed.; SpringerVerlag: Berlin, West Germany, 1984; pp 122130. (26) Fajer, J.; Brune, D. C.; Davis, M. S.; Forman, A,; Spaulding, L. C. Proc. Natl. Acad. Sei. U.S.A. 1975, 72, 49564960. (27) Fischer, S. F.; Scherer, P. 0. J . Chem. Phys. 1987, 115, 151158.
Warshel et al. region will be narrower than that shown for H123in Figure 3, because H I 2 ,is about half HI*. ( v ) Superexchange Involving Amino Acid Side Chains. Aromatic amino acids could contribute to electron transfer by a superexchangetype mixing of P* with C T states in which the amino acid is oxidized or reduced. A phenylalanine residue appears to provide a bridge for longrange electron transfer in cy* a tyrosine is located in close proximity to the tochrome c , ~ and chromophores in the reaction center.4 However, such mixing seems unlikely to be of major importance, because C T states in which tyrosine is oxidized or reduced are probably well above P* and P'B in energy. The midpoint potential for oxidation of the phenolic ring is approximately 1.45 V vs H'/H2,29 which is 1 .O V more positive than that for oxidation of P the midpoint potential for reduction of the phenolic ring is probably on the order of 3.1 V, more than 2 V more negative than that for the reduction of BC. If a CT state involving tyrosine is located 1.O eV (8066 cmI) above P* and is coupled to both P* and P'B by 20 cm', superexchange with this state would contribute a matrix element of only 0.05 cm' for electron transfer. This would augment the calculated matrix element for electron transfer from P* to B (H12) by less than 1%. Oxidation of the tyrosine could be made more favorable by ionization of the phenolic group, but it is questionable whether this would occur in the reaction center at physiological PH. The matrix element for the formation of a C T state in which an electron moves from the tyrosine to B is expected to be relatively small, because the C T state will mix only with the excited states of B and the tyrosine itself, not with the excited states of the special pair of BChls. As discussed above in connection with mechanism iv, the excited states B make only small contributions to P*. (vi) BackReactions. Matrix elements that connect P*, P'B, and the other C T states to the ground state are inclued in Figure 2. Like the matrix elements for the forward reactions, these coupling terms fall off rapidly as the distance between the electron donor and acceptor increases. The matrix element for the backreaction of P'B is calculated to be 8.0 cm', whereas that for P'H is 3.8 X lW5 cmI. The finding that the matrix element for the backreaction of P'B is comparable to, or even slightly larger than, that for the formation of this C T state from P* (5.9 cm') indicates, as has been suggested,I2 that the backreaction probably is not blocked by specific manipulation of the electronic coupling terms in the reaction center. The backreaction will be slowed down by unfavorable FranckCondon factors because of the large energy gap between the C T state and the ground state. Even so, efficient vibronic channels could leave it with a significant rate constantI2 so that the only way for the system to obtain a high quantum yield of charge separation may be to optimize the free energy barriers to make the forward reaction as fast as possible.'* When electron transfer from H to the quinone that serves as the next electron carrier is blocked, the P'H radical pair decays partly by a backreaction to the ground state and partly by evolving into a triplet state.7s30'32 At 295 K, the backreaction has a rate constant of approximately 5 X lo7 s'. The matrix element of 3.8 X cm' calculated for a direct backreaction of P'His much too small to account for this rate constant. One can, however, account for the observed kinetics by including superexchange through B'H. This state is connected to the ground state by a matrix element of 24 cmI (Figure 2). If B'H lies 1000 cmI above P+H, the matrix element for the decay of P+Hby superexchange would be approximately 0.3 cm' (24 X 11.1/1000), which is of the correct order of magnitude. Super
+
(28) Liang, N.; Pielak, G . J.; Mauk, A. G.; Smith, M.; Hoffman, B. M . 1987, Proc. Natl. Acad. Sei. U.S.A. 1987, 84, 12491252. (29) Vermillion, T. J.; Pearl, I. A. J Electrochem. SOC.1964, 111, 13921400. (30) Schenck, C. C.; Blankenship, R. E.; Parson, W. W Biochim. Biophys. Acta 1982, 680, 4459. (31) Boxer, S. G.; Chidsey, C. E. D.; Roelofs, M. G. Annu. Rec. Phys. Chem. 1983, 34, 389417. (32) Ogrodnik, A,; RemyRichter, N.; MichelBeyerle, M. E.; Feick, R. Chem. Phys. Lett. 1987, 135, 576581.
ElectronTransfer Pathways in Bacterial Photosynthesis exchange through P+B also could contribute to the decay, as Ogrodnik et al.32 have discussed recently.
Concluding Remarks Converting the structural information to electronic coupling provides an intriguingly complex picture of the initial steps of photosynthesis. It is clear that the distances between the neighboring edges of the chromophores are as important as the centertocenter distances in determining the electrontransfer kinetics. BChl and BPh are not structureless spheres or disks but rather are complex objects with substituent groups that participate in determining the coupling between the reaction center’s states. The exponential fall off of the overlap integrals between the a atomic orbitals on neighboring chromophores weights the interactions of close atomic pairs more heavily than interactions of atoms that are even a few angstroms farther apart. Because of the sensitivity of the matrix elements to the points of closest approach, movements of the chromophores could have major effects on the rates of electron transfer. Relaxations of P* could, for example, allow PLor P, to slide into a configuration that increases the coupling with B. The present work indicates that the ethylidine group of B could play particularly a important role in providing a “bridge” for electron transfer. In light of this finding, it is puzzling that the kinetics of the initial electrontransfer step are very similar in reaction centers from Rps. viridis and Rd. sphaeroides,”’O since the bacteriochlorophyll a that is present in Rd. sphaeroides has a saturated ethyl group in place of the ethylidine group. Because the distance between the nearest K atoms of P and B is considerably larger in Rd. sphaeroides than it is in Rps. viridis, a detailed analysis of the Rd. sphaeroides reaction center probably will have to include the nonconjugated ( a ) atoms of the chromophores, and possibly also the overlap with amino acid side chains. It also will be more sensitive to the longrange behavior of the atomic resonance integrals. Treatments of these effects will need to be calibrated carefully in simpler systems before they can be used with confidence. The electronic matrix elements that are important for charge separation in the reaction center include contributions from many exciton and C T states. We have considered five different mechanisms of charge separation, and it is possible that the actual process includes a combination of several of these pathways. However, the calculated coupling term for direct transfer of an electron from P* to H (mechanism ii) is much too small to account for the measured rate of charge separation. The coupling terms calculated for the superexchange mechanism (iii) also appear to be too small, unless the energy of P+B falls within a very narrow range. A similar conclusion has been reached recently by Marc ~who, estimated ~ ~ the matrix elements on the basis of magnetic (33) Marcus, R.Chem. Phys. Lerf. 1987, 133, 471477
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2701 field effects rather than structural information. The calculated rate constant for the pathway involving P+B as an intermediate (mechanism i) is of the right order of magnitude, but this pathway requires the intermediate state to be close to P* in energy, which implies a remarkable stabilization of P’B by the protein microenvironment. An alternative pathway involving B+H as an intermediate (iv) could be less problematic with regard to energetics but has somewhat smaller coupling terms. As we have emphasized above, however, the calculated matrix elements for all of the pathways are sensitive to small changes in the atomic positions. The relative merits of the different pathways will therefore need to be reevaluated after the crystallographic structure has been refined. The open question about the energy of the P’B C T state emphasizes the need to explore the role of the protein microenvironment in controlling charge separation in the reaction center.18,37 Specific stabilization of the C T states by protein dipoles could play a role in manipulatin thefree energies of the intermediates and in providing an optimal reorganization energy for the electron transfer reaction, thus ensuring that electron transfer is a rapid and essentially activationless process. As more details of the protein structure become available, it should be possible to develop correspondingly refined models of the chargetransfer process.
Acknowledgment. We thank Drs. J. Deisenhofer and H. Michel for providing the coordinates of the chromophores in the Rps. viridis reaction center (May 1985 data set), National Science Foundation Grants PCM8303385, PCM8312371, and PCM8616161 for support, and Digital Equipment Corp.’s ISIS program for help with the purchase of a computer. (34) We put the lowest energy transition of the form PM’PL (CT(PM+ PL)l in Table I) at 14400 cm’ and CT(P,+PM)l at 16400 cm’ (see ref 22). If these transitions are assumed to be degenerate (at 15 000 cmI), H I , decreases from 5.9 to 5.6 cm’; lH2,1 is essentially unchanged (15.0 cml). (35) Again, the coefficients depend on the energies of the CT transitions, but our assumptions on this point have only small effects on the calculated matrix elements for electron transfer. The values in Table I were obtained on the assumption that CT(PM+B)~and CT(PL+B)n are degenerate. If the transitions in which an electron moves from PLto B or H are put 2000 cml below the corresponding transitions in which the electron comes from PM,the coefficient for CT(PL+B)l in increases from 0.713 to 0.882, IHI21increases from 5.9 to 6.4 cml, and (H2,1 is essentially unchanged. (36) If the transitions in which PL transfers an electron to B or H are put 2000 cm’ below those in which the electron comes from PM,(Hilt(is essentially unchanged (2.5 cmI) and IHT31increases to 13.1 cmI. Omitting the ethylidine group of B has little effect on H12,but decreases IH24 to 0.6 cm’. (37) After this manuscript had been submitted we received the Xray coordinates of the reaction center of Rh. sph~eroides.~ These coordinates have been used in a free energy calculation that estimated the energetics of the chargetransfer states in the reaction center (Creighton, S.; Hwang, J.K.; Warshel, A,; Parson, W. W.; Norris, J. Biochemiszry 1988, 27, 774).
*,