Unified Theory on Rates for Electron Transfer Mediated by a Midway

J. Phys. Chem. B 2001, 105, 9603-9622

9603

Unified Theory on Rates for Electron Transfer Mediated by a Midway Molecule, Bridging between Superexchange and Sequential Processes Hitoshi Sumi*,† and Toshiaki Kakitani‡ Institute of Materials Science, UniVersity of Tsukuba, Tsukuba, 305-8573, Japan, and Department of Physics, Nagoya UniVersity, Chikusa-ku, Nagoya, 464-8602, Japan ReceiVed: January 4, 2001; In Final Form: May 3, 2001

A typical example of electron transfer (ET) mediated by a midway molecule M is the initial ultrafast ET from the special pair to bacteriopheophytin in the reaction center of bacterial photosynthesis, where the donor D and the acceptor A are so far apart (∼17 Å) that ET is mediated by a bacteriochlorophyll monomer located in-between. An analytic formula for the rate constant ka,d of such an ET is presented with attention to its morphology to the resonance Raman scattering in second-order optical processes. When M is located in the same energy region as D and A, important roles are played by the dephasing-thermalization time of phonons τm at M, relative to the lifetime of an electron lm at M. In the limit of τm . lm, the superexchange ET occurs where M mediates the ET as a virtual intermediate state of quantum mechanics, while in the opposite limit of τm , lm, the ordinary sequential ET occurs where ET to M from D is followed by ET to A from M after thermalization of phonons at M. The analytic formula correctly bridges the two limits. It describes intermediate cases as a single process, different from the expediency of assuming two channels by the superexchange and the ordinary sequential ET’s, which cannot coexist. Occurring earlier than τm in the course of ET are the superexchange ET and the subsequent hot sequential ET where ET to A from M occurs during reorganization of the medium around M after ET to M from D. Since they cannot be unambiguously separated, we can determine only the degree of ordinary sequentiality DOS of the ET, with DOS , 1 for the superexchange ET and 1 - DOS , 1 for the ordinary sequential ET. An analytic formula for DOS is also presented. DOS, in combination with ka,d, describes reasonably various aspects of the initial ET in bacterial photosynthesis, including its artificial modifications with respect to energy positions relative among D, M, and A.

I. Introduction Electron transfer (ET) reactions are the most fundamental of rate processes in condensed matter, being ubiquitous in a variety of phenomena in physics, chemistry, and biology.1 Therefore, formulation for their rate constants,2 pioneered by Marcus, has greatly promoted the understanding of various aspects of dynamical processes therein. Recently, a new frontier concerning ET mediated by a midway molecule has emerged as important especially in biological processes.3 A most typical example is the initial ET in the reaction center of bacterial photosynthesis, which is fastest in biological ET’s with a time constant ∼3 ps from the excited special pair to bacteriopheophytin at room temperature.4 The donor and the acceptor in the ET are so far apart, ∼17 Å center to center, that it is mediated by a bacteriochlorophyll monomer stationed between them. To cultivate the new frontier, it is requisite to have a convenient analytic formula for the rate constant of such ET’s. The purpose of the present work is to present it with an application to this typical example. This new frontier has emerged as an extension of the direct ET between the donor and the acceptor: For the comparison between theory and experiment to be as unambiguous as possible, the geometry of the donor-acceptor pair has often been artificially fixed by a rigid bridge of a spacer molecule in * Corresponding author. [email protected]. † University of Tsukuba. ‡ Nagoya University.

Fax:

+81

(298)

55-7440.

E-mail:

solvents.5 The spacer molecule is chosen so as not to be electronically inactive, not affecting the electronic states of the donor and the acceptor. For such a purpose, both the highest occupied and the lowest unoccupied electronic states of the spacer molecule are chosen so as to be well separated from the electronic states of the donor and the acceptor relevant to the ET between them. In this situation, the electronic states of the spacer molecule play a role of a quantum-mechanical virtual mediator of the ET. This mediation has been called the superexchange with a terminology originally used in magnetism6 where electron spins on two magnetic elements interact with each other through virtual mediation by electronic states on a nonmagnetic element stationed in-between. ET mediated by a long molecular chain in DNAs7 as well as in proteins8 proceeds as a sequence of such spacer-mediated elemental ET’s. In the initial ET in the reaction center of bacterial photosynthesis mentioned earlier, the special pair (P) of bacteriochlorophyll, a bacteriochlorophyll monomer (B), and bacteriopheophytin (H) are immersed as prosthetic groups in a transmembrane protein subunit which provides above-mentioned spacers for the ET. In this case, the intervening state of P+‚B-‚H is located in the same energy region as the donor state P*‚B ‚H and the acceptor state P+‚B‚H-. Therefore, it has been argued whether the state of P+‚B-‚H is a quantum-mechanical virtual mediator in the superexchange mechanism or a real intermediate station in the course of the long-ranged ET in the sequential mechanism.4 To clarify the nature of the initial ET, many investigations have been devoted also to artificial systems

10.1021/jp010018b CCC: $20.00 © 2001 American Chemical Society Published on Web 09/11/2001

9604 J. Phys. Chem. B, Vol. 105, No. 39, 2001 modified with respect to energy positions relative among the three states.4 After the ET, an electron is further transferred from thus reduced bacteriopheophytin to quinone in the trans-membrane protein subunit. It is also fast with a time constant of ∼200 ps at room temperature despite a large distance between them (∼14 Å center to center and ∼10 Å edge to edge). It has been considered here that a midway molecule facilitating this ET is installed as a tryptophane residue of the protein matrix.9 The analytic formula for the rate constant presented in the present work provides a convenient tool for clarifying in what situation such ET’s are governed by the superexchange or the sequential mechanism or must be regarded as a single intermediate process. Examples for calculations of the formula are shown, being taken from the initial ET in the reaction center of bacterial photosynthesis and its artificial modifications. II. Hot versus Ordinary Sequential ET The ET mediated by a midway molecule is theoretically isomorphic to second-order optical processes.10-12 In the latter, an initial state |i〉 is composed of a photon (of energy E) incoming to a matter in its electronic ground state, and a final state |f〉 is composed of a photon (of energy E′ < E) outgoing from the matter with a (usually, vibrational) excitation (with energy E - E′) left excited inside. Since there exists no direct coupling between these two states, they must be mediated by a midway state |m〉, which is given by an electronic excited state (of energy Em) of the matter without photon. When Em is much higher than E, |m〉 is passed as a virtual intermediate state of quantum mechanics. This situation is described as Raman scattering of the incident photon by the matter, corresponding to the superexchange ET in our original problem. When Em is located in the same energy region as E on a scale of the energy broadening of |m〉 due to interaction with phonons, the situation enters the resonance Raman scattering. It is known13,14 that in this situation a decisive role is played by the dephasingthermalization time τm of phonons at |m〉 and the lifetime of an electronic excitation lm at |m〉. When τm . lm, Raman scattering takes place since lm is so short that neither between |i〉 and |m〉 nor between |m〉 and |f〉, the quantum-mechanical coherence is disrupted in lm even if Em is located in the same energy region as E. When τm , lm, on the other hand, τm is so short that the coherence at |m〉 is quickly disrupted, with phonons quickly thermalized. In this case, the whole process can be divided into light absorption by the matter and subsequent luminescence from it after thermalization of phonons at |m〉. The luminescence in this case has been called the ordinary luminescence13,14 to distinguish it from the hot luminescence,15a which is emitted in the course of thermalization of phonons at |m〉. This case of consecutive absorption and ordinary luminescence corresponds, in our original problem, to a sequence of ET’s mediated by the midway state after thermalization of phonons there. Therefore, this situation can be called the ordinary sequential ET. When τm is comparable to lm in the situation of resonance Raman scattering, emitted here is not only the Raman scattering, but also the hot luminescence in the course of thermalization of phonons at |m〉. The former arises from the coherent part of the phonon scattering of an electron at |m〉 maintaining a phase memory with |i〉 or |f〉, while the latter arises from its incoherent part losing a phase memory.14,16 When the energy off-set between |m〉 and |i〉 is written as ∆E () Em - E), the former is emitted in the time period of ∼p/∆E, also in the ET mediated by a midway molecule.17 Even if ∆E is as small as the thermal

Sumi and Kakitani energy (∼200 cm-1) at room temperature, p/∆E is as small as ∼30 fs, being ∼0.1 times τm as shown later. At ∼p/∆E, however, the emitted light transforms gradually its feature from the former to the latter. Therefore, separation into the Raman scattering and the hot luminescence is conventional and theoretically cannot be unambiguously separated in general.13-16 It has been understood, therefore, that they are different features of a single process called the resonance Raman scattering. In our original problem of ET, taking place in this situation is not only the superexchange ET, but also a sequential ET occurring in the course of thermalization of phonons at |m〉. The latter can be called the hot sequential ET. Also in our original problem, when τm is comparable to lm, separation of ET into these two parts is conventional and ambiguous although the latter can be well-defined at time much later than that of ∼p/∆E. It is reasonable to understand analogously that they are different features of a single process called the ET mediated by a midway molecule. In conjunction with this, calculating the rate-constant formula of virtual mediation at the midway state for the superexchange ET, Kharkats et al.11 found that it gives rise also to a component describing a real transition there. This component is nothing but the hot sequential ET. The ET is essentially the same phenomena as the so-called hot transfer of an electron or excitation energy15b,18 triggered by an optical excitation of a donor in a short time during thermalization of phonons therein. With the understanding mentioned above in mind, let us look at the situation in the ET mediated by a midway molecule. For simplicity, we express hereafter the relevant three electronic states, i.e., the donor, the midway, and the acceptor states, by ket vectors |d〉, |m〉, and |a〉, respectively. It has been argued on the basis of numerical simulation that the discrimination between the superexchange and the (both hot and ordinary) sequential ET might always be determined by the energy position of |m〉, above or below |d〉.3a,19,20 An argument20 was also made that it is in conformity with our naive expectation if we could call ET mediated by |m〉 sequential when |m〉 is really occupied in the course of the ET, and superexchange otherwise. From the theoretical point of view, however, the discrimination is not so simple, especially in the intermediate situation where |m〉 is located in the same energy region as |d〉 or |a〉: Virtual mediation by |m〉 in the superexchange ET is equivalent to coherent mixing of |m〉 with either |d〉 or |a〉 in quantum mechanics. Therefore, the occupation extends also to |m〉 in the superexchange ET, and it becomes appreciable when |m〉 is in the same energy region as |d〉 or |a〉. Accordingly, a nonvanishing occupation at |m〉 does not distinguish the sequential ET, as well-known in second-order optical processes,13,14,16 although the opposite has been argued in terms of a long tradition in physical chemistry and biophysics.20 The superexchange and the sequential ET’s have been assigned to different groups of Feynman diagrams13,14 or pathways in the Liouville space10 for time evolution of the density matrix. Unfortunately, however, such separations are still conventional since partial rate constants obtained only by these groups of diagrams or pathways become negative for certain values of parameters, especially when τm is comparable to lm.10,13 When |m〉 is located in the same energy region as |d〉 or |a〉, the superexchange and the ordinary sequential ET’s become dominant for τm . lm and τm , lm, respectively. Therefore, they are concepts justifiable only in the mutually opposite limits, and they cannot coexist as two parallel channels. Nevertheless, however, the initial ET in the reaction center of photosynthesis has often been analyzed by an assumption that two channels

Electron Transfer Mediated by a Midway Molecule

J. Phys. Chem. B, Vol. 105, No. 39, 2001 9605

by the superexchange and the sequential ET’s coexist, and the branching ratio between them has also been argued.3a,20,21 Instead, we should consider reasonably that only the ordinary sequential ET can unambiguously be separated in the whole process of ET mediated by a midway molecule. An extent of its contribution to the whole rate constant of the ET was determined in ref 12, written as PSQ ( τjm (4.5)

f(τ) ≈ ka,mkm,d,

∫0τj

Pm ≡ exp[-

m

Cm(t) dt]

for τ > τjm

(4.8)

Equations 4.5 and 4.8 enable us to approximate the right-hand side of eq 4.1 by

ka,d ≈

∫0τj f(τ) exp[-∫0τ Cm(t) dt] dτ + k(OS) a,d Pm m

(4.9)

where k(OS) a,d ) ka,mkm,d/(ka,m + kd,m) represents the rate constant of ET in the ordinary sequential mechanism. Equality of this k(OS) a,d to that given by eq 3.2 is ensured by

Km ) exp[-∆Gm/(kBT)] ) km,d/kd,m

(4.10)

i.e., the principle of chemical equilibrium, which supplements eq 3.3. The second term on the right-hand side of eq 4.9 gives a contribution to ka,d of eq 4.1 from a τ region after τjm, where ET takes place after thermalization of phonons at |m〉, obeying the ordinary sequential mechanism. The first term, on the other hand, gives a contribution from the early τ region before τjm, where ET takes place either before or during thermalization of phonons at |m〉, obeying the superexchange or the hot sequential mechanism. Both terms are positive. As a degree of ordinary sequentiality in the ET mediated by a midway molecule, therefore, it seems reasonable to take

DOS ≡ (k(OS) a,d /ka,d)Pm,

with 0 < DOS < 1

(4.11)

The lifetime of an electron lm at |m〉 mentioned in section II can be determined by

exp[-

∫0l

m

Cm(t) dt] ) e-1

(4.12)

Since Pm of eq 4.6 represents the survival probability of an electron at |m〉 until after τjm (≈ 1.52 τm), it should be related to lm in eq 4.12 by

Pm , 1,

with

for τ > τjm (4.7)

for τm . lm, 1 - Pm , 1,

and for τm , lm (4.13)

(4.6)

which represents the survival probability of an electron at |m〉 until after thermalization of phonons therein. When τ . τm, the right-hand side of eq 4.3 approaches a value independent of τ, as shown in ref 12a. Considering eq 4.4, we interpret that this situation occurs for τ > τjm. With its explicit value12a independent of τ, therefore, we consider approximately

Since DOS of eq 4.11 is a product of two factors k(OS) a,d /ka,d and Pm, the superexchange ET for DOS , 1 can be realized typically in two different cases of k(OS) a,d /ka,d , 1 or Pm , 1. The former is realized when |m〉 is too high to be thermally accessible from |d〉. This is the usual static situation for the superexchange ET, since neither the ordinary nor the hot sequential ET can take place there. The latter is realized when τm . lm, as shown in eq 4.13, irrespective of the height of |m〉



relative to |d〉. This is the dynamical situation for the superexchange ET, since neither the ordinary nor the hot sequential ET can take place also here because of a very short lifetime lm of an electron while little dephasing among phonons at |m〉 occurs. When τm , lm, on the other hand, the hot sequential ET occurring in the course of thermalization of phonons at |m〉 can be neglected in comparison with the ordinary sequential ET, since Pm ≈ 1, as shown in eq 4.13. The superexchange ET is also negligible as long as |m〉 has an energy thermally accessible from |d〉, since it is weak in principle, being fourthorder in electronic transfer integrals, while the ordinary sequential mechanism is second-order. When τm , lm and simultaneously when |m〉 is located in the same energy region as |d〉, therefore, ET dominantly takes place by the ordinary sequential mechanism. In this case, in fact, DOS of eq 4.11 approaches unity since both k(OS) a,d /ka,d and Pm do so. In such ways, DOS of eq 4.11 describes correctly all the limiting situations in ET mediated by a midway molecule, as superexchange when DOS , 1 and ordinary sequential when 1 - DOS , 1. It should be noted, however, that DOS ) 1 does not necessarily mean ka,d ) (OS) k(OS) a,d , since eq 4.11 gives only ka,d < ka,d because of Pm < 1. This reflects a nonvanishing probability with which an electron returns back to |d〉 from |m〉 due to hot ET during thermalization of phonons at |m〉, not contributing to ka,d to |a〉.

respectively, relative to |d〉. Associated reorganization energies are given by

V. Rate Constant for ET Mediated by a Midway Molecule

Let 〈A〉0 represent the thermal average of an operator A on the Hamiltonian H0, similar to 〈A〉d in eq 3.6. Since qj and pj appear symmetrically in H0, we can obtain 〈qj2〉0 ) 1/2〈pj2 + qj2〉0 ) nj + 1/2, where nj ≡ 1/[exp(βpωj) - 1] represents the average excitation number of phonons with energy quantum pωj at temperature T. Therefore

Let us obtain, in the present section, an analytic formula for f(τ) of eq 4.2. Electronic states |d〉, |m〉, and |a〉 correspond to P*‚B‚H, P+‚B-‚H, and P+‚B‚H-, respectively, in the initial ET of bacterial photosynthesis. They are obtained by reducing P+, B, and H, respectively, in P+‚B‚H whose electronic state will be expressed as |0〉 hereafter. On reducing P+, an electron must be inserted to LUMO (i.e., lowest unoccupied electronic state) in P to obtain P*. Phonons in |0〉 are described by Hamiltonian H0. Let us express it as

H0 )

1

pωj(pj2 + qj2) ∑ 2 j

(5.1)

where qj and pj represent, respectively, the coordinate and its conjugate momentum operators for the jth normal mode with angular frequency ωj, being taken so as to satisfy

[pj,qk] () pjqk - qkpj) ) -iδj,k

(5.2)

Distortions of the medium will be described hereafter in reference to H0, unless otherwise is explicitly stated. In particular, Hd, Hm, and Ha in eq 3.4 are expressed as

Hd ) Hm ) Ha )

1

pωj[pj2 + (qj - x2ξj)2] ∑ 2 j

1

∑j pωj[pj2 + (qj - x2ηj)2] + ∆Gm

2

1

∑j pωj[pj

2

2

+ (qj - x2ζj) ] + ∆Ga 2

(5.3)

(5.4)

(5.5)

where ξj’s, ηj’s, and ζj’s describe distortions associated with electron occupation at |d〉, |m〉, and |a〉, respectively, while ∆Gm and ∆Ga give free energies after relaxation of |m〉 and |a〉,

λd )

∑k pωkξk2,

λm )

∑l pωlηl2

and λa )

pωmζm2 ∑ m

(5.6)

Since difference between Hd and H0 originates in the reduction of P+ in |0〉, it is localized only around P. Similarly, Hm - H0 and Ha - H0 are localized only around B and H, respectively. Since P, B, and H are apart enough from each other, it is reasonable to assume that Hd - H0, Hm - H0, and Ha - H0 do not overlap each other. This is to assume that ξk’s, ηl’s, and ζm’s in eq 5.6 do not overlap each other. The same assumption has been adopted also in ref 10. As quantities related to those in eq 5.6, we introduce also with β ) 1/(kBT),

Dd2 ≡

∑k (pωk)2ξk2 coth(βpωk/2)

(5.7)

Dm2 ≡

∑l (pωl)2ηl2 coth(βpωl /2)

(5.8)

(pωm)2ζm2 coth(βpωm/2) ∑ m

(5.9)

Da2 ≡

1 〈qj2〉0 ) coth(βpωj/2) 2

(5.10)

represents the average squared amplitude of each qj, including not only its thermal fluctuations at high temperatures but also its zero-point vibrations at low temperatures. In the semiclassical approximation which we take hereafter, each qj is regarded as distributed in proportion to exp(-1/2qj2/ 〈qj2〉0}).2 Since 〈(Hd - H0)2〉0 ) Dd2 + λd2, the electronic state |d〉 is located at an average energy λd with width Dd of eq 5.7 due to phonon fluctuations of the medium, seen from atomic and molecular configurations around the bottom of the Hamiltonian H0. Similarly, |m〉 and |a〉 are seen at ∆Gm + λm and ∆Ga + λa, respectively, with width Dm and Da of eqs 5.8 and 5.9. From the configurations around the bottom of the Hamiltonian Hd, on the other hand, |m〉 is seen to have an average energy

Emd ) ∆Gm + λm + λd

(5.11)

with width (Dm2 + Dd2)1/2. From the configurations around the bottom of the Hamiltonian Hm, similarly, |a〉 is seen with width (Da2 + Dm2)1/2 to have an average energy

Eam ) ∆Ga - ∆Gm + λa + λm

(5.12)

The semiclassical approximation is justified when both (Dm2 + Dd2)1/2 and (Da2 + Dm2)1/2 are much larger than the average energy quantum pω j of phonons interacting with an electron. In photosynthesis, pω j is determined by vibrations of the protein matrix around a pigment immersed therein, and they have average energies of about several tens of wave numbers, extending to about 100 cm-1.23 Moreover, the note mentioned

9608 J. Phys. Chem. B, Vol. 105, No. 39, 2001

Sumi and Kakitani with Eam of eq 5.12. Similar to sentences below eq 5.13, we understand that Eam represents the Franck-Condon energy from |m〉 to |a〉, i.e., the energy of |a〉 seen from |m〉 at the configuration of the medium relaxed under the condition that |m〉 is occupied. As shown also in Appendix B, f(τ) of eq 4.2 is given, in the same approximation, by

f(τ) ≈ JamJmd

( ) [ p

Figure 1. Adiabatic potential for the donor state |d〉 with bottom at O and that for the intermediate state |m〉 with bottom at M along the reaction coordinate Qmd for ET between them with the reorganization energy λm + λd and the free energy of reaction ∆Gm. Represented by Emd is the energy of |m〉 seen at a configuration of the medium at the bottom O of |d〉, and ∆Gmd* represents the thermal activation energy for ET from |d〉 to |m〉.

in the paragraph with eq 4.7 is justified under this semiclassical approximation, as explicitly shown in Appendix A. As shown in Appendix B, the semiclassical approximation reduces km,d of eq 3.6 to

(

) (

2

1/2

Jmd 2π km,d ≈ p D 2+D 2 m d

exp -

Emd2 2(Dm2 + Dd2)

)

(5.13)

ka,m ≈

(

2π

p D2+D 2 a m

)

1/2

e

-Sa

∞

∑

San

n ) 0 n!

(

exp -

∑

San

[(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4]1/2n)0 n! (npωa + Eam(τ) + Emd)2/2

Da + Dd + 2Dm - 2Dm(τ) 2

2

2

2

]

- y2 R[erf(x + iy)] (5.16)

with

[

]

(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4

x) 2

Da + Dd + 2Dm - 2Dm(τ) 2

2

2

2

1/2

τ p

(5.17)

y) {2[(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4][Da2 + Dd2 + 2Dm2 - 2Dm(τ)2]}1/2

(5.18)

(5.14)

When |a〉 is located as low as in the inverted region relative to |d〉 or |m〉, participation of an intramolecular mode with a large energy quantum pωa(. kBT) plays an important role in enhancing the rate of ET to |a〉.2 Such a phonon mode in the medium around |a〉 can easily be incorporated by shifting ∆Ga to ∆Ga + npωa in the expression of the rate constant for ET to |a〉 and simultaneously by averaging the rate constant with weight e-SaSan/n! for n ) 0, 1, 2, ..., where Sa is a coupling constant called the Huang-Rhys factor.2 With this expediency, ka,m is given, in the semiclassical approximation, by

Jam2

∞

2π exp(-Sa)

|[Dd2 + Dm2 - Dm(τ)2][npωa + Eam(τ)] - [Da2 + Dm2 - Dm(τ)2]Emd|

It is instructive to interpret this ET in Figure 1, since both the subsequent ET from |m〉 to |a〉 and the direct superexchange ET from |d〉 to |a〉 can be visualized in similar figures: At the configuration of the medium relaxed under the condition that |d〉 is occupied, the medium is distorted by energy λd only around the donor. At the configuration when |m〉 is occupied, the medium is distorted by energy λm only around the midway molecule. Measured from the latter configuration (at point M in Figure 1), the former (at point O in Figure 1) is distorted (along the coordinate Qmd in Figure 1) by energy λd around the donor and by energy λm around the midway molecule. Therefore, Emd of eq 5.11 gives the energy of |m〉 seen from |d〉 at the former configuration, i.e., seen vertically from point O in Figure 1. It has been called the Franck-Condon energy from |d〉 to |m〉 in spectroscopies. Similarly, the Franck-Condon energy Edm from |m〉 to |d〉 describes the energy of |d〉 seen from |m〉 at the latter configuration (i.e., seen vertically from point M in Figure 1), being given by

Edm ) -∆Gm + λm + λd

exp -

2

)

(npωa + Eam)2

2(Da2 + Dm2) (5.15)

Eam(τ) ) ∆Ga - ∆Gm + λa + λm - 2λm(τ)

∑l pωlηl2 cos(ωlτ)

(5.20)

∑l (pωlηl)2 coth(βpωl /2) cos(ωlτ)

(5.21)

λm(τ) ) Dm(τ)2 )

(5.19)

where R [erf(z)] represents taking the real part of the error function of a complex argument z defined24 by erf(z) ) (2/xπ)∫z0 exp(-t2) dt. Participation in the medium around |a〉 of an intramolecular mode with a large energy quantum pωa (.kBT) has been taken into account also in eq 5.16 with the same expediency as explained below eq 5.14. Since R [erf(x + iy)] {) 1 - R [erfc(x + iy)] by the complementary error function erfc(z)} for y > 0 appears also later in the formula for the rate constant k(SX) a,d for the superexchange ET, it seems convenient to show a method of its calculation as Appendix C. We should note here that λm(τ) and Dm(τ) of eqs 5.20 and 5.21 equal, respectively, λm of eq 5.6 and Dm of eq 5.8 at t ) 0, but both of them decay to zero as t increases because of cancellation among various cos(ωlτ)’s, i.e., dephasing among them.13,15 Their decay time can be estimated by the inverse of the average width in frequency dispersion of phonons contributing to the summation in eqs 5.20 and 5.21. Since it describes simultaneously thermalization of phonons at |m〉, it has been called the dephasing-thermalization time of phonons there, written as τm so far. In fact, λm - λm(τ), representing the average energy of medium reorganization around the midway molecule at time τ after ET to |m〉 from |d〉 at τ ) 0,13,15 is nearly equal to 0 for τ , τm and approaches λm for τ . τm. Here, λm(τ) itself represents the average energy of medium distortion at time τ measured from the configuration of the medium relaxed under the condition that |m〉 is occupied. Dm(τ) is concerned with the



width in energy fluctuations of |m〉 in the course of reorganization of the medium around the midway molecule,25 as shown in ref 13. To simplify the calculation of f(τ) of eq 5.16, both λm(τ) and Dm(τ)2 of eqs 5.20 and 5.21 are approximated: At high j , where pω j represents the average temperatures of kBT . pω energy quantum of phonons interacting with an electron, Dm(τ)2 reduces to 2kBTλm(τ). Replacing 2kBT therein by pω j coth(βpω j/ 2), which reduces correctly to 2kBT at high temperatures and behaves appropriately at low temperatures as eq 5.10, we approximate as

Dm(τ)2 ≈ pω j coth(βpω j /2)λm(τ)

(5.22)

Since Dm(0)2 ) Dm2 and λm(0) ) λm, this approximation naturally leads us also to

j coth(βpω j /2)λb, Db2 ≈ pω

for b ) a, m, and d (5.23)

The decay time of both λm(τ) and λd(τ) has been expressed as τm, where 1/τm can be estimated by the width in frequency dispersion of phonons interacting with an electron. Since λm(τ) is an even function of τ, it seems reasonable to adopt an approximation

λm(τ) ≈ λm exp(- τ2/τm2)

(5.24)

As mentioned in section IV, Cm(t) in eq 4.1 can approximately be given by the rate constant of decay by hot ET from |m〉 at time t after ET to |m〉 from |d〉 at t ) 0. The decay is composed of hot ET to |d〉 at time t, whose rate constant is written as kd,m(t), and that to |a〉, whose rate constant is written as ka,m(t). Accordingly, we write it as

Cm(t) ) kd,m(t) + ka,m(t)

(5.25)

As shown in Appendix D, kd,m(t) and ka,m(t) are given by

(

) (

Jmd2 2π kd,m(t) ) p D 2+D 2 d m

Edm(t)2

1/2

exp -

2(Dd2 + Dm2)

)

(5.26)

and

ka,m(t) ) Jam2

(

2π

)

1/2

p D2+D 2 a m

-Sa

e

∞

∑

San

n)0 n!

(

)

[npωa + Eam(t)]2

exp -

2(Da2 + Dm2) (5.27)

with

Edm(t) ) λd + λm - 2λd(t) - 2λm(t) - ∆Gm (5.28) and

λd(t) )

∑k pωkξk2 cos(ωkt),

[approximated as λd exp(- t2/τm2)] (5.29) where λd(t) is a quantity for the donor similar to λm(τ) of eq 5.20 for the midway molecule and can be approximated similarly. As clarified also in Appendix D, Edm(t) of eq 5.28 gives the energy of |d〉 seen from |m〉 at the average configuration of the medium partially relaxed around both the donor and the midway molecule, i.e., the effective Franck-Condon

energy from |m〉 to |d〉 at time t in the course of medium reorganization after ET to |m〉 from |d〉 at t ) 0. In fact, Edm(t) reduces to -Emd with Emd of eq 5.11 at t ) 0 since λm(0) ) λm and λd(0) ) λd, while it reduces to Edm of eq 5.14 at t . τm, where λm(t) ≈ λd(t) ≈ 0. Similarly, Eam(t) of eq 5.19, appearing in eq 5.27, represents the effective Franck-Condon energy from |m〉 to |a〉 at time t, reducing correctly to ∆Ga + λa - ∆Gm λm at τ ) 0 when |m〉 and |a〉 are at Emd of eq 5.11 and ∆Ga + λa + λd, respectively, while to Eam of eq 5.12 at τ . τm. At t . τm, kd,m(t) and ka,m(t) of eqs 5.26 and 5.27 approach correctly the correponding expression of the rate constants kd,m and ka,m (of eq 5.15) in the semiclassical approximation for ET from |m〉 to |d〉 and from |m〉 to |a〉 in thermal equilibrium of the medium. To estimate DOS, the right-hand side of eq 4.1 was approximated by eq 4.9. This estimation is based on an consideration that τjm defined by eq 4.4 can approximately be regarded as a boundary time much larger than τm. More accurate calculation for ka,d is given by enlarging the boundary time a bit more to 2τm since exp(- τ2/τm2) < 0.018 for τ > 2τm. In this case, the part obtained by integration from 2τm to ∞ in τ on the right-hand side of eq 4.1 for ka,d is approximated by (OS) m k(OS) exp[- ∫2τ represents the rate a,d 0 Cm(t) dt], where ka,d constant of the ordinary sequential ET given by eq 3.2. Therefore, eq 4.1 is approximated by

ka,d ≈

∫02τ

m

∫0τCm(t) dt] dτ + 2τ k(OS) a,d exp[-∫0 Cm(t) dt]

f(τ) exp[-

m

(5.30)

Thus, ka,d has turned out to be calculable by integrations of f(τ) of eq 5.16 with Cm(τ) of eq 5.25 in the time region from zero to 2τm. This is the analytic formula we propose for the rate constant of the ET mediated by a midway molecule, with approximations of eqs 5.24 and 5.29 for λm(τ) and λd(t) and of eqs 5.22 and 5.23 between λm(τ) and Dm(τ) et al. The degree of ordinary sequentiality DOS is given also analytically as eq 4.11 with eq 4.6. The later part of this section is devoted to verifying that the analytic formula of eq 5.30 (to be more exact, eq 4.1 with eq 5.16) for ka,d reduces to the rate constant k(OS) a,d of eq 3.2 for the ordinary sequential ET in the limit of τm , lm and to k(SX) a,d of eq 3.7 for the superexchange ET in the opposite limit of τm . lm, bridging the two limits in general cases. When τm , lm, the lifetime of an electron at |m〉 is so large in comparison with the thermalization time of phonons therein that both λm(τ) and Dm(τ) can be approximated as vanishing, since both of them approach zero for τ . τm. Cm(t) in eq 4.1 can be approximated by ka,m + kd,m of Cm(t) for t . τm, where ka,m + kd,m represents the rate constant for decay of an electron from |m〉 after thermalization of phonons at |m〉. These situations let ka,d of eq 4.1 with f(τ) of eq 5.16 reduce to k(OS) a,d of eq 3.2, as shown in Appendix E. When τm . lm, on the other hand, the lifetime of an electron at |m〉 is so short that λm(τ) and Dm(τ) can be approximated by λm and Dm, which λm(τ) and Dm(τ) approach, respectively, when τ , τm. Cm(t) in eq 4.1 can be approximated by Cm(0), which is written as 2Γm/p, representing the rate constant for decay of an electron from |m〉 in the medium thermally equilibrated under the condition that the electron still stays at |d〉. As shown in Appendix F, these situations let ka,d of eq 4.1 with f(τ) of eq 5.16 reduce to


k(SX) a,d

)

( )

1/2

2π

e

Da2 + Dd2

-Sa

∞

∑

San Jh(n)2 ad

n)0 n!

(

exp -

p

)

(npωa + Ead)2

2(Da2 + Dd2) (5.31)

with

Ead ) ∆Ga + λa + λd

(5.32)

where Ead gives the Franck-Condon energy from |d〉 to |a〉, obtained by changing “m” to “a” in eq 5.11. The right-hand side of eq 5.31 represents the rate constant for direct ET from |d〉 to |a〉, whose average energies are distributed at npωa + Ead for n ) 0, 1, 2, ... due to incorporation of an intramolecular vibrational mode, with an effective transfer integral Jh(n) ad dependent on n. Also, as shown in Appendix F, Jh(n) is given ad analytically by

( )

hJ(n)2 ad ) axπ with

a)

JamJmd 2 a2-b2 e R[ei2ab erfc(a + ib)] Γm

(

)

Da2 + Dd2 1 2 D 2D 2 + D 2D 2 + D 2D 2 m d d a a m

(5.33)

1/2

Γm

(5.34)

|Dd2[npωa + Eam(0)] - Da2Emd|

b)

(5.35)

[2(Dm2Dd2 + Dd2Da2 + Da2Dm2)(Da2 + Dd2)]1/2

Emd of eq 5.11 and Eam(0) ) ∆Ga - ∆Gm + λa - λm given by eq 5.19 at τ ) 0, where erfc(z) represents the complementary error function24 of a complex argument z defined by erfc(z) ) 1 - erf(z) with the error function erf(z). A method of calculating ea2R[ei2aberfc(a + ib)] is shown in Appendix C. The rate constant k(SX) h(n) a,d of eq 5.31 with J ad of eq 5.33 can be expressed by a form more straightforwardly intelligible as the rate constant of the superexchange ET, although it contains three integrations: As shown in Appendix G, eq 5.31 can be rewritten as

k(SX) a,d )

∫

n 2π -Sa ∞ Sa e p n)0 n!

dEm

x2πDm

∑ ∫

(

exp -

dEa

x2πDa

)

(Emd - Em)2 2

2Dm

|

(

exp -

) ( )

(npωa + Ead - Ea)2 2Da2

∫

dEd

JamJmd

|

x2πDd

Ea - Em + iΓm

exp -

Ed2

2

×

×

2Dd

2

δ(Ea - Ed) (5.36)

This equation means that energies Em and Ea of |m〉 and |a〉, respectively, are distributed around average energies Emd and npωa + Ead, with width Dm and Da in the medium thermally equilibrated under the condition that |d〉 is occupied, whose energy Ed is distributed around zero with width Dd. Superexchange ET from |d〉 to |a〉 takes place with energy conservation between Ed and Ea, virtually mediated by |m〉 with energy broadening Γm. Here, Emd and Ead, defined by eqs 5.11 and 5.32, respectively, represent the Franck-Condon energy from |d〉 to |m〉 and |a〉. Quantum-mechanical coherence among superexchange channels with different Em’s in eq 3.7 has been lost in eq 5.36 because of the semiclassical approximation for phonons

Sumi and Kakitani interacting with an electron at |m〉 in obtaining f(τ) of eq 5.16, since difference in Em in eq 3.7 arises from that in excitation numbers of such phonons. VI. Parameter Setting for the Initial ET in Bacterial Photosynthesis It will be demonstrated hereafter how appropriately the analytic formula eq 5.30 for the rate constant ka,d describes various situations in the ET mediated by a midway molecule. Preliminary calculation of eq 5.30 was briefly reported in ref 12b, taking as an example the initial ET in the reaction center of bacterial photosynthesis. Since it was found, however, that the calculation contains numerical errors, recalculation for the same example will be adopted here as a full report for demonstrating the applicability of eq 5.30. We had long-time controversies on free-energy values at |m〉 and |a〉, ∆Gm and ∆Ga in eqs 5.4 and 5.5, respectively, relative to that at |d〉. On the basis of recent measurements,27,28 it seems to have been being widely accepted that they are respectively about -450 and -2000 cm-1 for Rhodobacter (Rb.) sphaeroides. Therefore, let us take ∆Gm ) -450 cm-1 and ∆Ga ) -2000 cm-1 as a typical example in the present calculation. Since the reorganization energy associated with excitation of P to P* in the reaction center of bacterial photosynthesis was observed to be about 250 cm-1,23a the reorganization energy λd associated with the excitation reduction of P+ to P* should have a magnitude comparable to 250 cm-1. Let us take λd at 250 cm-1 also as a typical example. Both the midway molecule and the acceptor are a single pigment of B and H, respectively, in bacterial photosynthesis. The reorganization energy associated with the reduction of a single pigment is usually lager than associated with that of a dimer of pigments.29 Accordingly, both λm and λa associated with the reduction of B to B- and H to H-, respectively, should be a little larger than λa, which was taken above as 250 cm-1 associated with the reduction of a pair of bacteriochlorophylls comprising P. Let us take both λm and λa at 350 cm-1 as a typical example in the present calculation. In this case, the total reorganization energy associated with ET from |d〉 to |m〉 is given by λm + λd ) 600 cm-1. The thermal activation energy for the ET is given by the Marcus relation2

1 ∆Gmd* ) (∆Gm + λm + λd)2/(λm + λd) 4

(6.1)

This gives the energy of the transition state for the ET from |d〉 to |m〉, as shown in Figure 1. This energy turns out to be only about 9 cm-1. Therefore, λm ) 350 cm-1 corresponds to an assumption that ET from |d〉 to |m〉 is nearly barrierless. This situation is essential in reproducing the observed temperature dependence of the rate constant for ET from P*‚B‚H to P+‚B‚H-, which increases athermally with decreasing temperature.30 It has been taken into account in the formula eq 5.30 that the acceptor pigment distorts on ET to |a〉 also along the coordinate of an intramolecular vibration of an energy quantum pωa by the innersphere reorganization energy Sapωa, with a coupling constant of the Huang-Rhys factor Sa. The pωa contributing to the innersphere reorganization on ET has often been taken at 1500 cm-1.3,5,20,21 Let us take pωa at 1500 cm-1 as a typical example. Since Sa has usually a magnitude of the order of unity, let us take Sa at unity also as a typical example. These settings let the thermal activation energy for the unistep (superexchange) ET from |d〉 to |a〉 be very small, about 4 cm-1, calculated by


∆Gad* ) smallest of

(∆Ga + npωa + λa + λd)2 4(λa + λd) for n ) 0, 1, 2, ... (6.2)

as a modification of eq 6.1. This situation is also in tune with the general belief that biological processes have been adjusted so as to be most efficient in an environment in which they work, as a result of evolution over a long time. On the basis of this belief, the initial charge separation from |d〉 of P*‚B‚H to |a〉 of P+‚B‚H- in bacterial photosynthesis should have been adjusted so as to be fastest in order to avoid the loss of excitation in P* by fluorescent decay. To be so, this charge separation should be barrierless, located at the boundary between the normal and the inverted case from the standpoint of the direct ET.2 It will be shown later, however, that this situation is not essential in the initial charge separation in bacterial photosynthesis, where the energy position of the midway state |m〉 of P+‚B-‚H plays important roles in determining the whole rate constant. Correspondingly, the calculated rate constant of the initial charge separation in bacterial photosynthesis is not sensitive to such a choice of ∆Ga, λa, pωa, and Sa as adopted above. The thermal activation energy for ET from |m〉 to |a〉 is given by

∆Gam* ) smallest of


(∆Ga - ∆Gm + npωa + λa + λm)2 4(λa + λm) for n ) 0, 1, 2, ... (6.3)

It is 151 cm-1 in the present setting of parameters, having a value a little smaller than the thermal energy kBT (≈ 209 cm-1) at room temperature of 300 K. The thermalization time τm of phonons at |m〉 can be estimated by the inverse of the width in frequency dispersion of phonons interacting with an electron.13-15 The average energy quantum of these phonons has been denoted by pω j . In photosynthesis, a protein encompasses a pigment to and/or from which an electron is transferred. It has been observed that vibrations of a protein have in general average energies about several tens cm-1, extending to about 100 cm-1.23 As a typical example, therefore, let us take pω j at 50 cm-1 and τm at 0.3 ps, considering that the width in phonon-energy dispersion is about 100 cm-1. Such a rapid thermalization of protein distortions in the time region of 0.3 ps has in fact been observed in association with ET in a blue-copper protein, plastocyanin, by direct time-domain measurements (as ∼0.36 ps in this case).31 Most important in reproducing the observed value of the rate constant for ET from |d〉 of P*‚B‚H to |a〉 of P+‚B‚H- mediated by |m〉 of P+ ‚B- ‚H is the magnitude of the transfer integral Jmd for ET between |d〉 and |m〉. Its value can be obtained in association with the electronic-structure calculation of the protein-pigment complex comprising the reaction center. The rate constant at low temperatures (j 50 K) is very sensitive to the value of Jmd when the formula eq 5.30 is applied to the initial ET in bacterial photosynthesis. Unfortunately, however, its value scatters from 9 to 60 cm-1 in the INDO calculation32a,b and at 40 cm-1 in the extended Hu¨ckel calculation33 for Rhodopseudomonas (Rps.) Viridis. Inclusion of higher-order configuration interactions32c,34 still scatters Jmd from 9 to 35 cm-1. In this situation, taking average among 9, 40, and 60 cm-1 for Rps. Viridis, we set Jmd at 36 cm-1 as a typical value, considering also that this value is effective in reproducing well rate constants (1.4-1.3 × 1012 s-1)30 observed for Rps. Viridis at low temperatures. The rate constant at high temperatures

Figure 2. Temperature dependence of the rate constant for ET to |a〉 at -2000 cm-1 mediated by |m〉 at -450 cm-1 with Jmd ) 36 cm-1 and Jam ) 22 cm-1 or Jmd ) 24 cm-1 and Jam ) 27 cm-1, where Jmd and Jam represent, respectively, the electronic transfer integral between |d〉 and |m〉 and that between |m〉 and |a〉. Observed values of the rate constants reported in ref 30 for the initial ET in the reaction center of bacterial photosynthesis are shown by circles for Rps. Viridis and by rectangles for Rb. sphaeroides.

(J300 K) depends weakly on the transfer integral Jam for ET between |m〉 and |a〉 in the present calculation for the initial ET from |d〉 to |a〉 in bacterial photosynthesis. The rate constant observed for Rps. Viridis has a magnitude of about 3.7 × 1011 s-1 at room temperature.30 The calculated rate constant at room temperature was roughly tuned to this observed one through Jam. Its value thus set is 22 cm-1. Jmd ) 36 cm-1 is roughly in conformity with its value obtained by the extended Hu¨ckel calculation33 among 9, 40, and 60 cm-1. Jam ) 22 cm-1 is also roughly in conformity with this calculation, giving 20 cm-1. Bacteriochlorophylls in the reaction center are the b type in Rps. Viridis and the a type in Rb. sphaeroides.4 For Rb. sphaeroides, however, no intensive calculation for the electronic structure of the protein-pigment complex comprising the reaction center has been reported. Therefore, we have no reliable clue for estimating Jmd. In this situation for Rb. sphaeroides, we first set Jmd at 24 cm-1, with which we can reproduce roughly the observed rate constant (8.3-8.0 × 1011 s-1)30 at low temperatures (j50 K). Next, we set Jam at 27 cm-1, with which we can reproduce roughly the observed rate constant (∼3.5 × 1011 s-1)30 at room temperature, although its dependence on Jam values is weak also here. Figure 2 shows the temperature dependence of the rate constant calculated by eq 5.30 with the parameters set above, together with that30 observed for both Rps. Viridis (O) and Rb. sphaeroides (0). We see that the calculation reproduces well the observation as a whole. We should note here that it has been argued that these parameters change with temperature because of thermal expansion of the medium.35 If these parameters were known as a function of temperature, we should incorporate them in calculating the rate constant by eq 5.30. At present, however, it is not the case, and let us be content with the temperature-independent parameter setting, considering that at least it reproduces well the observed temperature dependence of the rate constant as shown in Figure 2.


Sumi and Kakitani

Figure 3. Temperature dependence of the degree of ordinary sequentiality for ET of Jmd ) 36 cm-1 in Figure 2, together with that of the rate constants calculated by an assumption of the ordinary sequential and the superexchange ET’s, and also that of the actual rate constant calculated by eq 5.30 for Rps. Viridis in Figure 2 for comparison.

VII. Mediation by |m〉 in the Same Energy Region as |d〉 and |a〉 It is the most characteristic feature of the rate constant for the initial ET in bacterial photosynthesis that it is not thermally activated, increasing monotonically as the temperature decreases, as seen in Figure 2. This fact indicates that the rate-limiting process is barrierless in this ET. Two nearly-barrierless processes are installed in the present parameter setting, i.e., ET from |d〉 to |m〉 with the thermal activation energy of ∼9 cm-1 and unistep (superexchange) ET from |d〉 to |a〉 with that of ∼4 cm-1, although ET from |m〉 to |a〉 with that of ∼151 cm-1 is not barrierless. Which barrierless process is rate limiting in bacterial photosynthesis? This question can be answered by looking at the degree of ordinary sequentiality, DOS of eq 4.11, associated with the ET. Shown by the dash-dot line in Figure 3 is the temperature dependence of DOS for the case of Jmd ) 36 cm-1 and Jam ) 22 cm-1, by which the ET rate constants observed for Rps. Viridis can well be reproduced as shown in Figure 2. The solid line in Figure 3 is the reproduction of that in Figure 2 for these parameter values. The dashed line in Figure 3 shows the rate constant k(OS) a,d of eq 3.2 for the ordinary sequential ET, while the dotted line therein shows that k(SX) a,d of eq 5.31 for the superexchange ET. DOS has a magnitude of 0.85-0.90 all through the whole temperature range below ∼300 K in Figure 3, and 1 - DOS is much smaller than unity. This means that the ET proceeds by the ordinary sequential mechanism where the contribution of the hot ET from |m〉 to |a〉 in the course of medium thermalization shortly after ET to |m〉 from |d〉 is small, as is the contribution of the unistep superexchange ET from |d〉 to |a〉. This is in conformity with the recent consensus3,27 that the initial ET in bacterial photosynthesis is mainly sequential. In fact, the rate constant shown by the solid line can roughly be approximated by the dashed line for k(OS) a,d , being much different from the dotted line for k(SX) . The a,d theoretical expression of k(OS) is given by eq 3.2, where Km a,d represents the equilibrium constant at |m〉 relative to |d〉, given by eq 3.3 with the free energy ∆Gm at |m〉 relative to |d〉. In the present parameter setting, ∆Gm is -450 cm-1, and Km is much

Figure 4. Temperature dependence of the actual rate constant and the degree of ordinary sequentiality for ET of Jmd ) 36 cm-1 in Figure 2 when τm was changed to 0.1, 0.6, and 2.0 ps from 0.3 ps therein, together with that of the rate constants calculated by an assumption of the ordinary sequential and the superexchange ET’s for comparison (although they are the same as those in Figure 2, being independent of τm).

larger than unity below ∼300 K in Figure 3. In eq 3.2, therefore, k(OS) a,d can be approximated by the rate constant km,d for ET to |m〉 from |d〉 as long as that ka,m for ET from |m〉 to |a〉 is not much smaller than km,d. The ET mediated by |m〉 is mainly limited by ET to |m〉 from |d〉 in the present parameter setting, in agreement with observation.27 Since ET from |d〉 to both |m〉 and |a〉 are nearly barrierless in the present parameter setting, |m〉 is located in the same energy region as |d〉 and also as |a〉, to which an electron is finally transferred. This energy structure is, however, not an origin of the ordinary sequential ET mentioned above, as shown below. In fact, the degree of ordinary sequentiality DOS can arbitrary be varied between zero and unity by changing artificially the dephasing-thermalization time τm of phonons at |m〉 without changing at all the static energy structure of the system. We cannot say, therefore, that the sequentiality of ET mediated by |m〉 is principally determined by the energy position of |m〉, such as sequential or superexchange when |m〉 is lower or higher than |d〉, respectively, although such an argument3,19,20 has been presented so far. As an explicit example, τm was changed to 0.1, 0.6, or 2.0 ps from the reasonable value of 0.3 ps in the system of Figure 3 with the static energy structure unchanged. Shown in Figure 4 is the temperature dependence of the rate constant (by the solid lines) and DOS (by the dash-dot lines) for τm ) 0.1, 0.6, (SX) and 2.0 ps. Both k(OS) a,d and ka,d do not depend on τm, which describes the speed of dynamics of the medium, with the same temperature dependence as shown by the dashed and the dotted lines in Figure 3. They are reproduced also in Figure 4. For τm ) 0.1 ps, the solid line is closer to the dashed line than that for τm ) 0.3 ps in Figure 3, and the dash-dot line with magnitude 0.95-0.97 is closer to unity than that for τm ) 0.3 ps in Figure


Figure 5. Temperature dependence of lm/τm for ET of τm ) 0.6 and 2.0 ps in Figure 4, together with that of the degree of ordinary sequentiality therein for convenience.

3. Accordingly, the ET becomes more ordinary sequential for τm ) 0.1 ps than for τm ) 0.3 ps. For τm ) 0.6 ps, on the contrary, the dash-dot line decreases to a magnitude of 0.700.77, and the solid line becomes about halfway between the (SX) dashed line of k(OS) a,d and the dotted line of ka,d . Consequently, the ET can be described neither as ordinary sequential nor superexchange, but only as a single process of the ET mediated by a midway molecule. For τm ) 2.0 ps, the solid line becomes nearly the same as the dotted line of k(SX) a,d , in correspondence with the fact that the dash-dot line of DOS decreases to a magnitude of 0.07-0.17. Accordingly, the ET becomes well describable by the superexchange mechanism despite the static energy structure which is the same as in Figure 3. It is interesting in the three examples mentioned above to look at the lifetime lm of an electron at |m〉, which is determined by eq 4.12. For both τm ) 0.1 and 0.3 ps, lm is much larger than τm. For τm ) 0.6 ps, the ratio of lm/τm decreases to a magnitude of 0.48-2.2 comparable to unity, as shown by the solid line in Figure 5, where DOS is also shown by the dashdot line for convenience. For τm ) 2.0 ps, the ratio decreases to a magnitude of 0.17-0.37, much smaller than unity, as shown by the corresponding solid line in Figure 5. We see, thus, that the ordinary-sequential ET for both τm ) 0.1 and 0.3 ps and the superexchange ET for τm ) 2.0 ps are in correspondence respectively to τm , lm and τm . lm in the present case, where |m〉 is located in the same energy region as |d〉 and |a〉, as pointed out in the last paragraph of section IV. VIII. Toward Mediation by |m〉 Much Higher than |d〉 and |a〉 By Jmd ) 24 cm-1 and Jam ) 27 cm-1, the ET rate constants observed for Rb. sphaeroides can well be reproduced, as shown in Figure 2. It can also be analyzed as in Figure 3 for Rps. Viridis. Figure 6 shows the temperature dependence of DOS by the dash-dot line in association with the rate constant shown by the solid line which reproduces that in Figure 2 for these parameter values. The dashed line in Figure 6 shows the rate constant k(OS) a,d of eq 3.2 for the ordinary sequential ET, while the dotted line therein shows that k(SX) of eq 5.31 for the a,d superexchange ET. DOS has a magnitude of 0.87-0.88 all


Figure 6. Temperature dependence of the degree of ordinary sequentiality for ET of Jmd ) 24 cm-1 in Figure 2, together with that of the rate constants calculated by an assumption of the ordinary sequential and the superexchange ET’s, and also that of the actual rate constant calculated by eq 5.30 for Rb. sphaeroides in Figure 2 for comparison.

through the whole temperature range in Figure 6, and 1 - DOS is much smaller than unity. This means that the ET proceeds also here by the ordinary sequential mechanism. In fact, the rate constant shown by the solid line can roughly be approximated by the dashed line for k(OS) a,d , being much different (OS) from the dotted line for k(SX) can be apa,d . In eq 3.2, ka,d proximated by km,d in the present parameter setting, and the ET mediated by the midway molecule is mainly limited by that from |d〉 to |m〉, also here with a situation similar to that in section VII for Rps. Viridis. The ordinary sequential ET mentioned above is realized as a result of dynamics, not as a result of the static energy structure, since the thermalization time τm () 0.3 ps) of phonons at |m〉 is much shorter than the lifetime lm of an electron at |m〉, also here in Rb. sphaeroides. In fact, by increasing τm, for example, to ∼2.5 ps also here, this ET can artificially be shifted to the superexchange ET, similar to the situation of τm ) 2.0 ps in Figure 4. The ordinary sequential ET mentioned above can be shifted to the superexchange ET also by changing the static energy structure, with τm kept unchanged here. Let us check this feature for the system of Jmd ) 24 cm-1 and Jam ) 27 cm-1 appropriate for Rb. sphaeroides, since relevant experiments3,4 have been performed mainly on this species. The most decisive role in this shift is played by the energy position ∆Gm of |m〉 relative to |d〉, which was set at -450 cm-1 in Figures 2 and 6. Modification in ∆Gm brings about a change in the thermal activation energy for ET both from |d〉 to |m〉 given by ∆Gmd* of eq 6.1 and from |m〉 to |a〉 given by ∆Gam* of eq 6.3, although that for unistep (superexchange) ET from |d〉 to |a〉 given by ∆Gad* of eq 6.2 is unchanged with a magnitude of 4 cm-1. Since the total reorganization energy associated with ET from |d〉 to |m〉 is given by λm + λd, set at 600 cm-1 in Figures 2 and 6, the ET becomes completely barrierless when ∆Gm is lowered to -600 cm-1, as understood from eq 6.1. We expect, therefore, that this change in ∆Gm will increase both the rate constant ka,d and the degree of ordinary sequentiality DOS. This change increases ∆Gam* a little to 175 cm-1 from 151 cm-1 in Figures 2 and 6. Shown by the solid


Figure 7. Temperature dependence of the actual rate constant and the degree of ordinary sequentiality for ET of Jmd ) 24 cm-1 in Figures 2 and 6 when ∆Gm was lowered to -600 cm-1 from -450 cm-1 therein, together with that of the rate constants calculated by an assumption of the ordinary sequential and the superexchange ET’s.

line in Figure 7 is the temperature dependence of ka,d, together (SX) with k(OS) a,d shown by the dashed line, and ka,d shown by the -1 dotted line, calculated for ∆Gm ) -600 cm . We see, in fact, that ka,d gets closer to k(OS) a,d in Figure 7 than in Figure 6 for ∆Gm ) -450 cm-1, with a concomitant larger deviation from k(SX) a,d . The magnitude of ka,d also increases in Figure 7 relative to that in Figure 6. The dash-dot line in Figure 7 represents the temperature dependence of DOS, with magnitude 0.89-0.92, becoming closer to unity than ∼0.87-0.88 for ∆Gm ) -450 cm-1 in Figure 6. With these changes, however, the temperature dependence of ka,d is qualitatively unaltered as a whole in Figure 7 from those in Figures 2 and 6. Let us change ∆Gm to -200 cm-1, increasing it by 250 cm-1 from -450 cm-1 in Figures 2 and 6. This change increases the thermal activation energy ∆Gmd* to 67 cm-1 from 9 cm-1 in Figures 2 and 6 and decreases that ∆Gam* to 57 cm-1 from 151 cm-1. The temperature dependence of the rate constant ka,d changes to that shown by the solid line in Figure 8, with k(OS) a,d changing to the dashed line and k(SX) a,d to the dotted line. We notice, first of all, that ka,d at low temperatures below ∼100 K increases with an increase in temperature in Figure 8 while decreases with temperature in Figures 2 and 6, although its temperature dependence above ∼100 K remains nearly unchanged, decreasing with it, together with a similar change in k(OS) a,d . The temperature dependence of ka,d deviates considerably from that of k(SX) a,d , reflecting that the degree of ordinary sequentiality shown by the dash-dot line in Figure 8 has a magnitude of 0.75-0.82. This means that the ET is mainly ordinary sequential all through the temperature range in Figure 8, although the superexchange ET from |d〉 to |a〉 is still nearly barrierless, with the thermal activation energy, ∆Gad* of eq 6.2, unchanged at 4 cm-1. Accordingly, the temperature-increasing dependence of ka,d at low temperatures, appearing newly here, arises from thermal activation over the barrier 67 cm-1 for ET from |d〉 to |m〉. The change in the temperature dependence of ka,d at ∼100 K mentioned above reflects that the thermal energy, amounting to ∼70 cm-1, exceeds 67 cm-1 there. Let us further increase ∆Gm to 500 cm-1 by 950 cm-1 from -450 cm-1 in Figures 2 and 6. This change increases the

Sumi and Kakitani

Figure 8. Temperature dependence of the actual rate constant and the degree of ordinary sequentiality for ET of Jmd ) 24 cm-1 in Figures 2 and 6 when ∆Gm was raised to -200 cm-1 from -450 cm-1 therein, together with that of the rate constants calculated by an assumption of the ordinary sequential and the superexchange ET’s.

Figure 9. Temperature dependence of the actual rate constant and the degree of ordinary sequentiality for ET of Jmd ) 24 cm-1 in Figures 2 and 6 when ∆Gm was raised to 500 cm-1 from -450 cm-1 therein, together with that of the rate constants calculated by an assumption of the ordinary sequential and the superexchange ET’s.

thermal activation energy ∆Gmd* to 504 cm-1 from 9 cm-1 in Figures 2 and 6 although decreases that ∆Gam* to 32 cm-1 from 151 cm-1. The temperature dependence of the rate constant ka,d changes to that shown by the solid line in Figure 9, with k(OS) a,d changing to the dashed line and k(SX) a,d to the dotted line. We notice here that the temperature dependence of ka,d becomes completely reversed from that in Figures 2 and 6, always increasing with increasing temperature. At a glance, it looks like the typical temperature dependence of the rate constant for ET between two molecules. In fact, the rate constant of the twocenter ET typically becomes nearly temperature-independent at low temperatures below ∼100 K due to phonon tunneling through a nonvanishing reaction barrier produced by medium


Figure 10. Temperature dependence of the actual rate constant and the degree of ordinary sequentiality for ET of Jmd ) 24 cm-1 in Figures 2 and 6 when ∆Gm was raised to 1500 cm-1 from -450 cm-1 therein, together with that of the rate constants calculated by an assumption of the ordinary sequential and the superexchange ET’s.

distortion, while at high temperatures it becomes thermally activated during fluctuations of the medium for surmounting the barrier.2 ET underlining the ka,d with the temperature dependence in Figure 9 is, however, qualitatively different from the usual one between two molecules: Although above ∼150 K the solid line for ka,d approaches the dashed line for k(OS) a,d , (SX) below ∼75 K, however, it approaches the dotted line for ka,d . Since ∆Gmd* is considerably larger than kBT below ∼300 K, making the rate constant km,d for ET from |d〉 to |m〉 small enough, k(OS) a,d given by eq 3.2 can be approximated by km,d. Therefore, the ka,d above ∼150 K originates in ET from |d〉 to |m〉 with a thermally activated temperature dependence, while below ∼75 K it arises from unistep superexchange ET from |d〉 to |a〉 whose thermal activation energy ∆Gad* nearly vanishes with a magnitude of 4 cm-1. This is also reflected in the temperature dependence of the degree of ordinary sequentiality DOS shown by the dash-dot line in Figure 9. It has a magnitude of 0.60-0.72 above ∼150 K, while it is smaller than 0.07 below ∼75 K. We have thus obtained that the superexchange ET below ∼75 K switches to the ordinary-sequential ET above ∼150 K in this single system as the temperature increases. Let us finally increase ∆Gm to 1500 cm-1, by 1950 cm-1 from -450 cm-1 in Figures 2 and 6. This change increases the thermal activation energy ∆Gmd* very much to 1800 cm-1 from 9 cm-1 in Figures 2 and 6, although decreases that ∆Gam* to 14 cm-1 from 151 cm-1. The temperature dependence of the rate constant ka,d changes to that shown by the solid line in (SX) Figure 10, with k(OS) a,d changing to the dashed line and ka,d to the dotted line. DOS shown by the dash-dot line is ∼0.48 at ∼300 K and decreases rapidly toward zero below ∼300 K. We see, accordingly, that the superexchange ET appearing newly in Figure 9 below ∼75 K extends to the whole temperature region below room temperature in Figure 10. Below ∼300 K, in fact, the solid line merges swiftly to the dotted line for k(SX) a,d . It becomes increasing with decreasing temperature. This athermal temperature dependence reflects that the thermal activation energy ∆Gad* for the superexchange ET nearly vanishes with a magnitude of 4 cm-1. This temperature dependence also clearly tells us that it does not originate in usual ET between


Figure 11. Temperature dependence of the actual rate constant and the degree of ordinary sequentiality for ET of Jmd ) 24 cm-1 in Figures 2 and 6 when ∆Ga was raised to 0 cm-1, i.e., to the same energy as that of |d〉, from -2000 cm-1 therein, together with that of the rate constants calculated by an assumption of the ordinary sequential and the superexchange ET’s, and also that of the rate constant for ET of ∆Ga ) -2000 cm-1 therein for comparison.

two molecules, whose rate constant becomes only nearly temperature-independent at low temperatures typically below ∼100 K due to phonon tunneling through a nonvanishing reaction barrier.2 We note also that the solid line deviates markedly from the dashed line for k(OS) a,d below ∼300 K. Modification in the energy of |d〉 has effects similar to that in ∆Gm investigated above, since the characteristics of the ET from |d〉 to |a〉 mediated by |m〉 is dominantly determined only by the relative energy structure between |d〉 and |m〉 in the present parameter setting in Figures 2, 3, and 6. This means also that no drastic change in the rate constant ka,d is expected by modification in the free energy of reaction ∆Ga as long as ∆Ga < 0 and |∆Ga| . kBT, with τm ) 0.3 ps, as in Figures 2-10: In this case, ET is nearly ordinary sequential, and its rate constant can be approximated by kOS a,d of eq 3.2. In eq 3.2, Km given by eq 3.3 is much larger than unity below ∼300 K when ∆Gm ) -450 cm-1 in Figures 2-10. Therefore, kOS a,d is nearly unaltered by a change in ∆Ga as long as ka,mKm . km,d, being approximated by km,d, which is independent of ∆Ga. For the system of Figure 6, this feature was checked by changing ∆Ga to -1000 and -3000 cm-1 from -2000 cm-1 in Figure 6, although figures are not shown explicitly. ka,d is virtually unaltered by such changes in ∆Ga in the temperature range of Figure 6. DOS is also virtually unaltered for ∆Ga ) -1000 cm-1 from that in Figure 6, being larger than ∼0.85, and it becomes even larger for ∆Ga ) -3000 cm-1 than that in Figure 6, becoming larger than ∼0.91. Accordingly, the ET remains nearly ordinary sequential with such changes in ∆Ga, reflecting that k(OS) a,d can roughly be estimated by km,d in such cases. When ∆Ga is further raised to realize a situation where Kmka,m is not much larger than km,d, the rate constant ka,d will be influenced by such a change in ∆Ga. This takes place when the condition of ∆Ga < 0 and |∆Ga| . kBT is not satisfied. As an example, ∆Ga was raised to zero from its value of -2000 cm-1 in Figure 6, obtaining Figure 11 where the temperature (SX) dependence of ka,d, k(OS) a,d , ka,d , and DOS are shown by the solid, the dashed, the dotted, and the dash-dot lines, respectively.

9616 J. Phys. Chem. B, Vol. 105, No. 39, 2001 The rate constant in the original case of ∆Ga ) -2000 cm-1 in Figure 6 is also shown by the thin solid line for comparison. We see that the ET remains nearly ordinary sequential with DOS > 0.88 below room temperature, and ka,d can nearly be approximated by k(OS) a,d except low temperatures below ∼50 K. Although ka,d is nearly unaltered from that for ∆Ga ) -2000 cm-1 below ∼50 K, it deviates markedly from that above ∼ 50 K, becoming about one-third the value at room temperature. This shows that (Kmka,m)-1 cannot be neglected in comparison with (km,d)-1 in eq 3.2 for k(OS) above ∼50 K, although ka,d ad (OS) approaches ka,d there. IX. Discussion Recently, Michel-Beyerle and her collaborators36 succeeded in raising ∆Gm by ∼1000 cm-1, replacing B with vinyl B in the reaction center of Rb. sphaeroides.27(d) They observed that it completely reversed the temperature dependence of the rate constant of ET from P* to H from the temperature-decreasing one of the wild type shown in Figure 2: It becomes nearly independent of temperature below ∼120 K while increasing with increasing temperature above ∼120 K. This observation enabled them to conclude that they succeeded in observing a transition of the ET mediated by a midway molecule from the superexchange mechanism in the low-temperature region to the ordinary sequential mechanism in the high-temperature region in a single system. Their observation is just in conformity with the result of the present calculation shown in Figure 10. It was reported in ref 37 that a rise of the free energy of reaction ∆Ga by ∼1000 cm-1 by replacement of H with pheophytin in the reaction center of Rb. sphaeroides decreased the rate constant ka,d to about half at room temperature. In the present calculation, when ∆Ga is raised to zero, ka,d decreases to about one-third its value in the wild type at room temperature, as shown in Figure 11. Here, ka,d is not sensitive to the variation in ∆Ga as long as -3000 cm-1 j ∆Ga j -1000 cm-1, as noted in the second lowest paragraph of section VIII. Moreover, the detailed figure of ∆Ga ≈ -2000 cm-1 adopted in Figures 2, 3, and 6 has not fully been settled yet for the reaction of the wild type. Accordingly, we infer that the replacement mentioned above corresponds to a situation that ∆Ga was raised to a value between -500 cm-1 and zero, probably in the upper half of the region. Allen, Williams and collaborators found that the potential for oxidation of P to P+ in the reaction center of Rb. sphaeroides can be modified widely from 95 mV below to 260 mV above that of the wild type by changing the number of hydrogen bonds to P between 0 and 4 from 1 in the wild type.38 The observed rate constant of ET from |d〉 to |a〉 changes continuously with the change in the oxidation potential. When it is changed to 260 mV above that of the wild type, the rate constant at roomtemperature becomes ∼1/14 times as small as that in the wild type.38(b) It was also observed that the peak energy of optical absorption of P is not sensitive to such changes in the oxidation potential.38c This means that the potential for oxidation of P* to P+ also shifts parallel to that of P to P+. Let us note here that ∆Gm gives the free energy of |m〉 with electronic configuration P+‚B-‚H measured from |d〉 with that P*‚B‚H. If the interaction of B- with P+ was kept unchanged with the modification in the oxidation potential of P/P+, then ∆Gm would shift in a range of 766 cm-1 () 95 meV) below that of the wild type to 2100 cm-1 () 260 meV) above it since the energy for oxidation of P*‚B‚H to P+‚B‚H shifts just antiparallel to this shift. The free energy of reaction ∆Ga also would shift by

Sumi and Kakitani the same amount as ∆Gm. Theoretically, the energy position of |m〉 is very decisive in regulating the rate constant ka,d, as shown in Figures 7-10. Such a big modification in ∆Gm would result in a very big change in ka,d. As mentioned above, however, even a modification in the oxidation potential as large as 260 mV above that of the wild type decreases ka,d only to a magnitude of one-tenth. It is suspected, therefore, that the interaction of both B- and H- with P+ may also be changed with the modification in the oxidation potential. It has been reported,39 in fact, that the charge distribution on the two bacteriochlorophylls in P+ is altered parallel to the change in the number of hydrogen bonds to P introduced for modification of its oxidation potential. A change in character between the superexchange and the ordinary sequential ET’s has often been analyzed so far on the basis of an assumption that there coexist two parallel channels for the superexchange and the ordinary sequential ET’s, and the branching ratio between them has been discussed.3,20,21 Theoretically, they are concepts justifiable only in mutually opposite limits of τm . lm and τm , lm, respectively, as explicitly demonstrated in Figures 4 and 5. Also, the situation in Figure 9 for τm ) 0.3 ps with a switch between them at ∼100 K can be shifted either to the superexchange ET in the whole temperature range of Figure 9 by artificially increasing τm or to the ordinary sequential ET there by decreasing τm, without changing the static energy structure of the system parametrized by ∆Gm, ∆Ga, λm, λa, Jmd, and Jam. It is meaningless, in principle, to assume that both the superexchange ET realized for τm/lm . 1 and the ordinary sequential ET for τm/lm , 1 coexist as two parallel channels in a single system with a definite value of τm/lm. ET with such a change in character should be described as neither superexchange nor sequential in terms of the old concepts for ET between two molecules. We are looking at a unified single process, new in itself, called the ET mediated by a midway molecule in parallel to the resonance Raman scattering in second-order optical processes. X. Conclusion The present work is devoted, first of all, to presenting an analytic formula, eq 5.30, for the rate constant ka,d of the ET mediated by a midway molecule as a single process. A measure DOS for the degree of ordinary sequentiality of the ET is also presented as an analytic formula, eq 4.11. When the midway state |m〉 is located in the same energy region as the donor state |d〉 and the acceptor state |a〉, a decisive role is played by the dephasing-thermalization time τm of phonons at |m〉 relative to the lifetime of an electron lm there. In the limit of τm . lm, the ET proceeds as superexchange, where |m〉 is passed as a virtual intermediate state in quantum mechanics. In the opposite limit of τm , lm, the ET proceeds as ordinary sequential, where the ET to |m〉 from |d〉 is followed by that to |a〉 after thermalization of phonons at |m〉. In intermediate cases between the two limits, there occurs also the hot sequential ET, where the second ET to |a〉 takes place during thermalization of phonons at |m〉. Both ka,d and DOS describe reasonably various aspects of the ET, in their explicit calculation for the initial ET in the reaction center of bacterial photosynthesis and also as predictions concerning the change in the energy of |m〉 in Figures 7-11.40 Acknowledgment. The authors would like to thank Professor J. Jortner of Tel Aviv University, Israel for valuable and stimulating discussions.



1 〈eiHaσ/pe-iH0σ/p〉0 ≈ exp i(∆Ga + λa)σ/p - Da2σ2/p2 2

Appendix A: Decay Time of the Two Terms in Equation 4.7

[

Since |m〉 is seen with width (Dm2 + Dd2)1/2 from configurations around the bottom of the Hamiltonian Hd, the integrand in eq 3.6, equal to the second term on the right-hand side of eq 4.7, should have a decay time in |µ| of the order of p/(Dm2 + Dd2)1/2. Similarly, the first term therein should have a decay time in |σ| on the order of p/(Da2 + Dm2)1/2. Since both (Dm2 + Dd2)1/2 and (Da2 + Dm2)1/2 have been assumed to be much larger than the average energy quantum of phonons interacting with an electron in the semiclassical approximation, these decay times are much smaller than the dephasing-thermalization time of such phonons τm, which has a magnitude on order the typical oscillation period of them. Appendix B: Derivation of Eq 5.16 in the Semiclassical Approximation When Hd - H0 does not overlap with Hm - H0, the integrand in eq 3.6 can be decomposed as

〈eiHmµ/pe-iHdµ/p〉d ) 〈eiHmµ/pe-iH0µ/p〉0〈eiH0µ/pe-iHdµ/p〉d (B.1) where 〈A〉0 represents the thermal average of an operator A on the phonon Hamiltonian H0, being defined similarly to 〈A〉d. The first term on the right-hand side of eq B.1 is concerned with phonons appearing only in Hm - H0, and they are averaged on H0 since such phonons are in thermal equilibrium on H0 when |d〉 is occupied by an electron on the left-hand side of eq B.1. Similarly, W(µ,σ;τ) of eq 4.3 can be decomposed into

W(µ,σ;τ) ) 〈eiHaσ/pe-iH0σ/p〉0〈eiH0µ/pe-iHdµ/p〉dM(µ,σ;τ) (B.2) with

M(µ,σ;τ) ≡ 〈eiHm(τ+µ/2-σ/2)/peiH0σ/pe-iHm(τ-µ/2+σ/2)/pe-iH0µ/p〉0 (B.3) The semiclassical approximation in the paragraph with eq 5.11 allows us to expand the logarithm of each bracketed term appearing on the right-hand side of eqs B.1 and B.2 up to second-order in µ and σ.2 For example, using 〈Hm - H0〉0 ) λm + ∆Gm with λm defined by eq 5.6, the first bracketed term on the right-hand side of eq B.1 can be approximated as

[

〈eiHmµ/pe-iH0µ/p〉0 ≈ exp i(λm + ∆Gm)µ/p 1 〈(H - H0 - λm - ∆Gm)2〉0µ2/p2 2 m 1 ) exp i(λm + ∆Gm)µ/p - Dm2µ2/p2 2

[

]

]

(B.6)

The second term is the same as eq B.5, and the third term, M(µ,σ;τ) of eq B.3, reduces to

[

M(µ,σ;τ) ≈ exp i(∆Gm + λm)(µ - σ)/p + 1 2i(λm - λm(τ))σ/p - Dm2(µ2 + σ2)/p2 + Dm(τ)2µσ/p2 2 (B.7)

]

with λm(τ) of eq 5.20 and Dm(τ) of eq 5.21, as derived by using the next four paragraphs. To digest the right-hand side of eq B.3, let us first introduce a unitary operator U for coordinate shift in the phonon space by

U ) eiP,

with P ) x2

∑l ηl pl

(B.8)

The commutation relation of eq 5.2 ensures that it shifts the coordinate operator ql to

UqlU-1 ) ql + x2ηl

(B.9)

although it does not shift the momentum operator pl. These properties ensure that it converts any analytic function of the phonon Hamiltonian Hm of eq 5.4, say g(Hm), to

Ug(Hm)U-1 ) g(H0 + ∆Gm)

(B.10)

with H0 of eq 5.1. Next, we introduce a Heisenberg operator of P in eq B.8 at time t by

P(t) ≡ exp(iH0t/p)P exp(-iH0t/p) )

x2∑ηl [pl

cos(ωl t) - ql sin(ωl t)] (B.11)

l

where the quantity in the square bracket on the right-hand side represents the time evolution of pl which has the same form in quantum mechanics as in classical mechanics. To apply the unitary transformation of eq B.10, let us rewrite M(µ,σ;τ) of eq B.3 into

M(µ,σ;τ) ) 〈U-1UeiHm(τ+µ/2-σ/2)/pU-1UeiH0σ/pU-1 Ue-iHm(τ-µ/2+σ/2)/pU-1Ue-iH0µ/p〉0 (B.12)

] (B.4)

Applying eq B.10 to eq B.12 and then using eq B.11 with eq B.8, we get

where the second equality can be derived by using eqs 5.8 and 5.10. Similarly, the second bracketed term on the right-hand side of eq B.1 can be approximated as

M(µ,σ;τ) ) ei∆Gm(µ-σ)/p〈e-iPeiP(τ+µ/2-σ/2)e-iP(τ+µ/2+σ/2)eiP(µ)〉0 (B.13)

1 〈eiH0µ/pe-iHdµ/p〉d ≈ exp -iλdµ/p - Dd2µ2/p2 2

When a commutator [A,B] between operators A and B is commutable with both A and B, an operator equality eAeB ) exp(A + B + (1/2)[A,B]) holds.26 A pair of P(t) and P(t′) satisfy this condition, the commutator between them being calculated by eq 5.2. This enables us to combine the product of four exponentials in the bracket on the right-hand side of eq B.13 into a single exponential. To be explicit, the bracketed term therein can be cast into

[

]

(B.5)

Using eqs B.4 and B.5 for eq B.1, we can perform the µ integration in eq 3.6 to get eq 5.13. The semiclassical approximation allows us to expand also each term in eq B.2 with respect to µ and σ up to second-order in its logarithm. The first term in eq B.2 reduces to


{ [(

exp i

∑l ηl2 2 sin ωl

µ-σ 2

sin ωl µ + sin ωlσ

- sin ωl

]}〈 [ (

) ( )

µ+σ 2

iP τ +

2

with

cos ωl τ +

exp - iP + iP τ + µ+σ

Sumi and Kakitani

)

µ-σ 2

+ iP(µ)

]〉

a ≡ Emd + pω , -

(

σ µ-σ µ+σ + iλm - i2λm(τ) 〈eiQ〉0 p p p (B.15)

)

∑l ωl ηl [pl (σ sin ωl τ) - ql (µ - σ cos ωl τ)]

Q ) x2

(B.16)

with the use of λm of eq 5.6 and λm(τ) of eq 5.20. Since Q of eq B.16 is linear in both µ and σ with vanishing average value, 〈eiQ〉0 ) exp(-1/2〈Q2〉0) holds, and it reduces to

∑l ωl 2ηl 2[〈pl 2〉0(σ sin ωl τ)2 + 〈ql 〉0(µ - σ cos ωl τ) ]} (B.17) 2

2

Both 〈pl 2〉0 and 〈ql 2〉0 equal the right-hand side of eq 5.10 from the symmetry in the Hamiltonian H0 of eq 5.1. Insertion of them into eq B.17 allows us to express the summation in l therein by Dm2 of eq 5.8 and Dm(τ)2 of eq 5.21, and we see that eq B.15 equals eq B.7. Equations B.5-B.7 for W(µ,σ;τ) of eq B.2 enable us to perform analytically the µ and σ integration for f(τ) in eq 4.2, as shown by using the next three paragraphs, to arrive at eq 5.16. A function of x which is unity for |x| < 2τ, but zero otherwise can be expressed by

1 H(x) ) 2πi

∫-∞ ∞

[

]

eiω(x+2τ) eiω(x - 2τ) dω ω - iδ ω - iδ 1 ∞ sin(2ωτ) iωx e dω (B.18) ) -∞ π ω

with a positive infinitesimal δ. H(µ - σ) expressed above let us rewrite f(τ) of eq 4.2 as

( )∫

1 JamJmd π p

2

∞

-∞

sin(2ωτ) g(ω;τ) dω ω

(B.19)

with

∫-∞ dµ∫-∞ dσ e

1 g(ω;τ) ) 2 p

∞

∞

iω(µ-σ)

2

2

W(µ,σ;τ) (B.20)

Inserting W(µ,σ;τ) of eq B.2 with eqs B.5-B.7, we can cast g(ω;τ) of eq B.20 into

(B.21)

-1

∫

[

Dm2 + Dd2 2 dσ ‚‚‚ exp µ -∞ p 2p2

∫

∞

dµ -∞ p

∞

Da + D m 2

2

2p2

σ2 +

]

Dm(τ)2 µσ (B.24) p2

where Emd and Eam(τ) are given by eqs 5.11 and 5.19, respectively. Since F(τ) of eq B.23 equals the integration in µ and σ of the exponential factor on the right-hand side of eq B.24, 〈‚‚‚〉 therein defines an average in µ and σ. On this definition, the average of aµ/p + bσ/p, linear in µ and σ, vanishes, and eq B.21 reduces to

1 g(ω;τ) ) F(τ) exp - 〈(aµ/p + bσ/p)2〉 2

[

) F(τ) exp -

[

]

]

(Da2 + Dm2)a2 + 2Dm(τ)2ab + (Dm2 + Dd2)b2

2[(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4] (B.25)

since the logarithm of the right-hand side of eq B.21 is quadratic in µ and σ. Inserting eq B.22, we can express the right-hand side of eq B.25 as a function of ω, as

{

g(ω,τ) ) F(τ) exp -

{

exp -

}

[Eam(τ) + Emd]2 1 × 2 D 2 + 2D 2 + D 2 - 2D (τ)2 a m d m

1 Da + 2Dm + Dd - 2Dm(τ) [c(τ)Emd 2 (D 2 + D 2)(D 2 + D 2) - D (τ)4 2

a

2

m

2

m

2

d

}

m

cj(τ)Eam (τ) - pω]2

(B.26)

with

c(τ) ≡

Da2 + Dm2 - Dm (τ)2

, Da2 + 2Dm2 + Dd2 - 2Dm (τ)2 and cj(τ) ≡ 1 - c(τ) (B.27)

Since 1/(ω + iR) ) - iπδ(ω) + P (1/ω) for R ) +0, where P represents taking the principal part of the integration, sin(2ωτ)/ω in the integrand on the right-hand side of eq B.19 for f(τ) can be rewritten into πδ(ω) + uei2ωτ/(ω + iR), where u represents taking the imaginary part. Therefore, the ω integration in eq B.19 can be cast into a form as

∫

g(ω;τ) ) F(τ)〈exp[iaµ/p + ibσ/p]〉

(B.23)

〈‚‚‚〉 ≡ F(τ)

∫

f(τ) )

2π [(Da + Dm )(Dm + Dd2) - Dm (τ)4]1/2 2

and

with

〈eiQ〉0 ) exp{-

(B.22)

(B.14)

0

Obeying the semiclassical approximation, let us expand the logarithm of eq B.14 in both µ and σ up to the second order. The exponent in the bracketed term in eq B.14 is a linear combination of pl ’s and ql ’s with vanishing average value. Therefore, it is sufficient to expand the exponent up to first order in both µ and σ, and eq B.13 reduces to

M(µ,σ;τ) ≈ exp i∆Gm

F(τ) ≡

b ≡ Eam(τ) - pω

[

]

{

(pω - A)2 sin(2ωτ) 2 dω ) e-A /D π + exp ω 2D (E - A - iDτ/p)2 (A + iDτ/p)2 dE u exp + E + iR D D (B.28)

∫

[

]}

for D > 0, after a variable change from ω to E ) pω. We note here a functional relation24

Electron Transfer Mediated by a Midway Molecule (-y+ix)2

e


e-t dt ) -iπ erfc(x + iy) -∞-y + ix - t

∫

∞

2

(B.29)

R erf(x + iy) ) erf(x) +

2

2

2

2

sin (xy) + e π 2

-x2

2

πx

∑

2

2

e R [e

where R represents taking the real part and the equality is derived from the definition of erfc(z) ) 1 - erf(z) from the error function erf(z), together with R [erf(x + iy)] ) R [erf(x - iy)]. Since the left-hand side of eq B.28 is equal to eq B.30, f(τ) of eq B.19 reduces to

f(τ) )

2π(JamJmd/p)2 [(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4]1/2 (Eam(τ) + Emd)2/2

Da2 + Dd2 + 2Dm2 - 2Dm(τ)2 with

[

]

- y2 R [erf(x + iy)] (B.31)

]

(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4

x) 2

[

exp -

Da2 + Dd2 + 2Dm2 - 2Dm(τ)2

1/2

τ p

(B.32)

y) |[Dd2 + Dm2 - Dm(τ)2]Eam(τ) - [Da2 + Dm2 - Dm(τ)2]Emd| {2[(Da2 + Dm2)(Dm2 + Dd2) - Dm(τ)4][Da2 + Dd2 + 2Dm2 - 2Dm(τ)2]}1/2

(B.33) As explained in the paragraph with eq 5.15, we here apply to f(τ) of eq B.31 the expediency for incorporating an intramolecular phonon of a large energy quantum pωa with the coupling constant of the Huang-Rhys factor Sa at the acceptor. We thus obtain eq 5.16. Appendix C: Real Part of the Complementary Error Function of a Complex Argument The error function of a complex argument can be expressed by a functional series24

erf(x + iy) ) erf(x) + [1 - cos(2xy) + i sin(2xy)]e-x /(2πx) 2 2 ∞ 2 [fm(x,y) + igm(x,y)]e-m /4/(m2 + 4x2) (C.1) + e-x π m)1 2

∑

2

2

This can be applied for the numerical calculation of eq 5.16. For that of eq 5.33, noting erfc(a + ib) ) 1 - erf(a + ib), we can use a2

and y ) -A/xD

2

m)1x

with

x ) xDτ/p,

e-m /4

∞

+ m2/4 {[cosh (my/2) sin (xy) - sinh (my/2) cos (xy)] x + m cosh(my/2) sinh(my/2) cos(xy) sin(xy)} (C.4)

for x > 0, where erfc(z) represents the complementary error function of a complex argument z. Therefore, the right-hand side of eq B.28 reduces to

e-y {π + u [-iπ erfc(x + iy)]} ) πe-y R [erf(x + i|y|)] (B.30)

e-x

i2ab

a2

erfc(a + ib)] ) cos(2ab)e erfc(a) + 2a

e-m /4

∞

sin2(ab)

+

πa

2

∑ π m)1

[sin2(ab) + sinh2(mb/2)] (C.5)

a + m /4 2

2

Appendix D: Derivation of Equations 5.26 and 5.27 At time t after ET to |m〉 from |d〉 at t ) 0, the medium is partially reorganized around both the donor and the midway molecule unless t is much larger than the thermalization time τm of phonons. The medium is distorted, on the average, by energy λm(t) around the midway molecule, measured from the configuration relaxed under the condition that |m〉 is occupied. Measured from the configuration relaxed under the condition that |d〉 is occupied, on the other hand, the medium is distorted by energy λm - λm(t) around the midway molecule. Similar interpretation can be given to λd(t) of eq 5.29 with respect to the medium distortion around the donor. In this case, however, λd - λd(t) represents the average energy of medium distortion around the donor at time t measured from the configuration relaxed under the condition that |d〉 is occupied, with λd - λd(t) growing as t increases. At time t, the medium is distorted, on the average, by energy λd - λd(t) around the donor and by λm - λm(t) around the midway molecule, measured from the configuration relaxed under the condition that |d〉 is occupied. Considering in parallel to the paragraph with eq 5.13 and Figure 1, therefore, we understand that, at the configuration partially reorganized at time t, |d〉 is seen, on the average, to have a total energy

Ed(t) ) λm - λm(t) + λd - λd(t)

(D.1)

λd(t) itself represents the average energy of medium distortion around the donor at time t, measured from the configuration relaxed under the condition that |m〉 is occupied, with λd(t) decreasing as t increases. From the configuration relaxed under the condition that |a〉 is occupied, finally, the medium at time t is distorted, on the average, by energy λd(t) around the donor, by λm - λm(t) around the midway molecule, and by λa around the acceptor. We see, accordingly, that, at the configuration partially reorganized at time t, |m〉 and |a〉 are seen, respectively, at total energies

with

fm(x,y) ) 2x - 2x cosh(my) cos(2xy) + m sinh(my) sin(2xy) (C.2) and

gm(x,y) ) 2x cosh(my) sin(2xy) + m sinh(my) cos(2xy) (C.3) Inserting eqs C.2 and C.3 into eq C.1, we get the real part of erf(x + iy), expressed as

Em(t) ) ∆Gm + λm(t) + λd(t)

(D.2)

Ea(t) ) ∆Ga + λa + λm - λm(t) + λd(t)

(D.3)

and

Since Eam(t) of eq 5.19 and Edm(t) of eq 5.28 equal, respectively, Ea(t) - Em(t) and Ed(t) - Em(t), they give the energies of |a〉 and |d〉 seen from |m〉 at this average configuration of the medium, i.e., the effective Franck-Condon energies from |m〉


Sumi and Kakitani

to |a〉 and |d〉, respectively, at time t in the course of medium reorganization after ET to |m〉 from |d〉 at t ) 0. km,d of eq 5.13 gives the rate constant for ET from |d〉 to |m〉 in the medium thermally equilibrated under the condition that |d〉 is occupied. Its exponential factor is determined by the Franck-Condon energy Emd of eq 5.14 from |d〉 to |m〉, together with the width (Dm2 + Dd2)1/2 in thermal-equilibrium fluctuations of the energy of |m〉 seen from |d〉 around Emd. At time t in the course of medium reorganization after ET to |m〉 from |d〉 at t ) 0, Edm(t) of eq 5.28 gives the effective Franck-Condon energy from |m〉 to |d〉. The energy of |d〉 seen from |m〉 fluctuates around Edm(τ). The width of the fluctuation is always maintained at its thermalequilibrium value (Dm2 + Dd2)1/2 in such a linear system as described by Hamiltonians in eqs 5.1 to 5.5, due to ergodicity, since the medium was fluctuating in thermal equilibrium at t ) 0 when the reorganization started. (This situation is in contrast with that noted in ref 25 where the configuration of the medium started to reorganize from a single value on its coordinate at t ) 0.) Accordingly, the rate constant kdm(t) of hot ET from |m〉 to |d〉 at time t is given by eq 5.26. Similarly, Eam(t) of eq 5.19 gives the effective Franck-Condon energy from |m〉 to |d〉 at time t. The energy of |a〉 seen from |m〉 at time t fluctuates around this average value with the same width (Da2 + Dm2)1/2 as that in thermal equilibrium because of ergodicity. Accordingly, the rate constant kam(t) of hot ET from |m〉 to |a〉 at time t is given by eq 5.27.

situation, [sin(2ωτ)]/ω in the ω integration for f(τ) of eq B.19 can be approximated by πδ(ω) since |sin(2ωτ)/ω| is much ∞ smaller than its value at ω ) 0 for ω . τ-1, together with ∫-∞ dω [sin(2ωτ)]/ω ) π. Accordingly, f(τ) can be approximated by (JamJmd/p)2g(0;∞). It is equal to ka,mkm,d, as shown also in eq 4.8, with the rate constant km,d of eq 5.13 for ET from |d〉 to |m〉 and that ka,m of eq 5.15 for ET from |m〉 to |a〉, when we recover the expediency for incorporating an intramolecular phonon of a large energy quantum pωa with the coupling constant of the Huang-Rhys factor Sa at |a〉. Since the typical value of t in exp[-∫τ0Cm(t) dt] has a magnitude of order lm from eq 4.12, it can be regarded as much larger than τm when τm , lm. In this t region, Cm(t) approaches ka,m + kd,m, with the rate constant kd,m for ET from |m〉 to |d〉, as mentioned in sentences above eq 4.4. Therefore, we can approximate exp[-∫τ0Cm(t) dt] by exp[-(ka,m + kd,m)τ]. Together with f(τ) ≈ ka,mkm,d for τ . τm, this enables us to see that eq 4.1 for ka,d can be approximated by ka,mkm,d/(ka,m + kd,m) when τm , lm. It can be rewritten into k(OS) a,d of eq 3.2 with the use of the principle of chemical equilibrium of eq 4.10. Although f(τ) ≈ ka,mkm,d for τ . τm was derived above in the semiclassical approximation, it can be justified without this approximation, as verified in ref 12a. Therefore, ka,d ≈ k(OS) a,d for τm , lm holds without this approximation.

Appendix E: Derivation of Equation 3.2 from Equation 5.16 in the Small τm Limit

The rate constant ka,d is expressed by eq 4.1 as an integration of f(τ) exp[-∫τ0Cm(t) dt] in τ, where f(τ) of eq 5.16 is equivalent to that of eq B.19 with g(ω;τ) of eq B.26 and F(τ) of eq B.23 when we neglect the expediency for incorporating an intramolecular phonon of a large energy quantum pωa at the acceptor. As mentioned in the first paragraph of Appendix E, the dominant contribution to the integration in eq 4.1 is given by τ with a magnitude on the order of lm. When τm . lm, the τ is much smaller than τm. In this τ region, Dm(τ) and Eam(τ) in both eqs B.23 and B.26, defined by eqs 5.21 and 5.19, respectively, remain at their initial value Dm(0) ( ) Dm) and Eam(0) ( ) ∆Ga + λa - ∆Gm - λm). On the other hand, exp[∫τ0Cm(t) dt] can be approximated by exp(-2Γm τ/p) when C(0) is written as 2Γm/p. Therefore, in the τ integration in eq 4.1, there remains that of [sin(2ωτ)/ω] exp[-2Γm τ/p], and a change of variable from ω to E ≡ pω in g(ω;τ) of eq B.26 casts eq 4.1 into

The rate constant ka,d is expressed by eq 4.1 as an integration of f(τ) exp[-∫τ0Cm(t) dt] in τ. Here, f(τ) of eq 5.16 is equivalent to that of eq B.19 with g(ω;τ) of eq B.25 when we neglect the expediency for incorporating intramolecular phonons of a large energy quantum pωa at the acceptor. f(τ) approaches a constant when τ is much larger than τm which represents the thermalization time of phonons at |m〉. From eq 4.12, exp[-∫τ0Cm(t) dt] decays exponentially to zero when τ is much larger than lm which represents the lifetime of an electron at |m〉. Therefore, the dominant contribution to the integration in eq 4.1 for ka,d is given by τ with a magnitude of order of lm. When τm , lm, the τ is much larger than τm. In this τ region, Dm(τ) in eq B.25, defined by eq 5.21, decays to zero, and Eam(τ) in eq B.22, defined by eq 5.19, approaches Eam of eq 5.12. Thus, g(ω;τ) of eq B.25 with eq B.22 reduces to

[

g(ω;∞) ) F(∞) exp -

(Emd + pω)2 2(Dm + Dd ) 2

2

-

(Eam - pω)2

]

2(Da + Dm ) (E.1) 2

2

with

Appendix F: Derivation of Equation 5.31 from Equation 5.16 in the Large τm Limit

ka,d ) F with

F≡ 2π F(∞) ) 2 2 1/2 (Da + Dm ) (Dm2 + Dd2)1/2

(E.2)

g(ω;∞) as a function of pω has a width of order (Dm2 + Da2)1/2 or (Da2 + Dm2)1/2 around its peaks. In sentences below eq 5.12, both (Dm2 + Da2)1/2 and (Da2 + Dm2)1/2 were assumed to be much larger than the average energy quantum pω j of phonons interacting with an electron. The thermalization time τm of phonons at |m〉 has a magnitude on the order of 1/ω j . We can consider, therefore, that the typical value of ω in g(ω;∞) is much larger than 1/τm. Since the typical value of τ is much larger than τm, that of ωτ is much larger than unity. In this

(

∫-∞∞

exp[-B(E - A)2]

(JamJmd)2 2B 2 p Da + Dd2

E2 + Γm2

) ( 1/2

exp -

dE

Ead2 2(Da2 + Dd2)

(F.1)

)

1 B ≡ (Da2 + Dd2)/(Dm2Dd2 + Dd2Da2 + Da2Dm2) 2

(F.2)

(F.3)

and

A ≡ [Da2Emd - Dd2Eam(0)]/(Da2 + Dd2)

(F.4)

The E integration in eq F.1 is equivalent to

1 u Γm

-B(E-A)2

-t2

∫-∞∞eE - iΓm dE ) Γ1mu∫-∞∞- b -e ia - t dt

(F.5)



with

a ≡ ΓmxB,

and b ≡ -AxB

(F.6)

where u represents taking the imaginary part. Here, the functional relation of eq B.29 enables us to rewrite the righthand side of eq F.5 into

1 u Γm

[

-t2

dt ∫-∞∞- b-e - ia - t

]

/

πea -b R [ei2ab erfc(a + ib)] Γm (F.7) 2

)

2

where b on the right-hand side can be replaced by |b|. Eq F.5 with eq F.7 cast eq F.1 into

ka,d )

( )(

πa JamJmd p Γm

2

2 2 Da + Dd2

) ( ) 1/2

exp -

Ead2/2 Da2 + Dd2

+

a2 - b2 R [ei2ab erfc(a + ib)] (F.8) with

a)

[

(Da2 + Dd2)/2

Dm2Dd2 + Dd2Da2 + Da2Dm2

]

1/2

Γm

(F.9)

and

b)

|Dd2Eam(0) - Da2Emd| [2(Dm2Dd2 + Dd2Da2 + Da2Dm2)(Da2 + Dd2)]1/2

(F.10)

As explained in the paragraph with eq 5.15, we can here apply to ka,d of eq F.8 the expediency for incorporating intramolecular phonons of a large energy quantum pωa with the coupling constant of the Huang-Rhys factor Sa at the acceptor. We thus arrive at the expression of the rate constant k(SX) a,d of eq 5.31 with Jh(n) of eq 5.33. ad Appendix G: Derivation of Equation 5.36 To derive eq 5.36 from eq 5.31 is equivalent to deriving eq 5.31 from eq 5.36 by one-to-one correspondence. Introducing an integration in E in the integrand of eq 5.36 by inserting ∫ dE δ(Em - Ed - E), we can perform the integrations in Ed, Em, and Ea therein. As a result, we are led to eq E.1, neglecting the expediency for incorporating intramolecular phonons of a large energy quantum pωa with the coupling constant of the Huang-Rhys factor Sa at the acceptor. Equation E.1 is equivalent to eq 5.31, as shown in Appendix F. References and Notes (1) Review articles: Balzani, V. Electron Transfer in Chemistry; WileyVCH: Weinheim, 2001; Vol. 1-5. (2) A review article: Sumi, H. In Electron Transfer in Chemistry Balzani, V., Ed.; Wiley-VCH: Weinheim, 2001; Vol. 1 (Principles, Theories, Techniques and Methods), p 64. (3) Review articles: (a) Bixon, M.; Jortner, J. AdV. Chem. Phys. 1999, 106, 35. (b) Page, C. C.; Moser, C. C.; Chen, X.; Dutton, P. L. Nature 1999, 402, 47. (4) Review articles: (a) Blankenship, R. E.; Madigan, M. T.; Bauer, C. E. Anoxygenic Photosynthetic Bacteria; Kluwer: Dordrecht, 1995. (b) Hoff, A. J.; Deisenhofer, J. Phys. Rep. 1997, 287, 1. (5) For example, Closs, G. L.; Miller, J. R. Science 1988, 240, 440. (6) Kramers, H. A. Physica 1934, 1, 182. (7) (a) Barbara, P.; Olson, E. J. J. AdV. Chem. Phys. 1999, 106, 647. (b) Bixon, M.; Giese, B.; Wessely, S.; Langenbacher, T.; Michel-Beyerle, M. E.; Jortner, J. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 11713.

(8) (a) Skourtis, S. S.; Beratan, D. N. AdV. Chem. Phys. 1999, 106, 377. (b) Regan, J. J.; Onuchic, J. N. AdV. Chem. Phys. 1999, 108, 497. (c) Kawatsu, T.; Kakitani, T.; Yamato, T. Inorg. Chim. Acta 2000, 300-302, 862. (9) Plato, M.; Michel-Beyerle, M. E.; Bixon, M.; Jortner, J. FEBS Lett. 1989, 249, 70. (10) (a) Hu, Y.; Mukamel, S. Chem. Phys. Lett. 1989, 160, 410; J. Chem. Phys. 1989, 91, 6973. (b) Ku¨hn, O.; Rupasov, V.; Mukamel, S. J. Chem. Phys. 1996, 104, 5821. (11) Kharkats, Y. I.; Kuznetsov, A. M.; Ulstrup, J. J. Phys. Chem. 1995, 99, 13545. (12) (a) Sumi, H.; Kakitani, T. Chem. Phys. Lett. 1996, 252, 85. (b) A preliminary report of the present formulation. Sumi, H. J. Electroanal. Chem. 1997, 438, 11. (13) (a) Toyozawa, Y. J. Phys. Soc. Jpn. 1976, 41, 400. (b) Toyozawa, Y.; Kotani, A.; Sumi, A. J. Phys. Soc. Jpn. 1977, 42, 1495. (14) Kotani, A.; Toyozawa, Y. J. Phys. Soc. Jpn. 1976, 41, 1699. (15) (a) Hizhnyakov, V.; Tehver, I. Phys. Status Solidi 1967, 21, 755; 1970, 39, 67. (b) SoV. Phys. JETP 1975, 42, 305. (16) See also, Kurita, H.; Sakai, O.; Kotani, A. J. Phys. Soc. Jpn. 1980, 49, 1920. (17) Nitzan, A.; Jortner, J.; Wilkie, J.; Burin, A. L.; Ratner, M. A. J. Phys. Chem. B 2000, 104, 5661. (18) Sumi, H. J. Phys. Soc. Jpn. 1982, 51, 1745; Phys. ReV. Lett. 1983, 50, 1709. (19) Egger, R.; Mak, C. H. J. Phys. Chem. 1994, 98, 9903. (20) Bixon, M.; Jortner, J. J. Chem. Phys. 1997, 107, 5154. (21) (a) Bixon, M.; Jortner, J.; Michel-Beyerle, M. E. Biochim. Biophys. Acta 1991, 1056, 301. (b) Chan, C.-K.; DiMagno, T. J.; Chen, L. X.-Q.; Norris, J. R.; Fleming, G. R. Biophysics 1991, 88, 11202. (c) Joseph, J. S.; Bialek, W. J. Phys. Chem. 1993, 97, 3245. (d) Nagarajan, V.; Parson, W. W.; Davis, D.; Schenck, C. C. Biochemistry 1993, 32, 12324. (e) Bixon, M.; Jortner, J.; Michel-Beyerle, M. E. Chem. Phys. 1995, 197, 389. (f) Zusman, L. D.; Beratan, D. N. J. Chem. Phys. 1999, 110, 10468. (g) Okada, A.; Bandyopadhay, T. J. Chem. Phys. 1999, 111, 1137. (22) Kimura, A.; Kakitani, T. Chem. Phys. Lett. 1998, 298, 241. (23) (a) Lyle, P. A.; Kolaczkowski, S. V.; Small, G. L. J. Phys. Chem. 1993, 97, 6924. (b) Ahn, J. S.; Kanematsu, Y.; Enomoto, M.; Kushida, T. Chem. Phys. Lett. 1995, 215, 336. (24) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover: New York, 1965. (25) To be more exact, {[Dm4 - Dm(τ)4]1/2/Dm gives the width of the fluctuations at time t under the condition that the configuration of the medium, described by x2∑jpωjηjqj in eq 5.4, is put at a certain unrelaxed point at τ ) 0 with ET to |m〉. The width starts from zero at τ ) 0 and approaches the thermal equilibrium value Dm as τ increases beyond τm. (26) For example, see: Messiah, A. Me´ canique Quantique; Dunod; Paris, 1959. (27) (a) Schmidt, S.; Arlt, T.; Hamm, P.; Huber, H.; Na¨gele, T.; Wachtveitl, J.; Meyer, M.; Scheer, H.; Zinth, W. Chem. Phys. Lett. 1994, 223, 116. (b) Schmidt, S.; Arlt, T.; Hamm, P.; Huber, H.; Na¨gele, T.; Wachtveitl, J.; Zinth, W.; Meyer, M.; Scheer, H. Spectrochim. Acta, Part A 1995, 51, 1565. (c) Huber, H.; Meyer, M.; Scheer, H.; Zinth, W.; Wachtveitl, J. Photosynth. Res. 1998, 55, 153. (d) Spo¨rlein, S.; Zinth, W.; Meyer, M.; Scheer, H.; Wachtveitl, J. Chem. Phys. Lett. 2000, 322, 454. (28) Ogrodnik, A.; Keupp, W.; Volk, M.; Aumeier, G.; Michel-Beyerle, M. E. J. Phys. Chem. 1994, 98, 3432. (29) Toyozawa, Y. In Dynamical Processes in Solid State Optics; Kubo, R., Kamimura, H., Eds.; Shokabo: Tokyo, 1967; p 90. (30) Fleming, G. R.; Martin, J.-L.; Breton, J. Nature 1988, 333, 190. (31) Nakashima, S.; Seike, K.; Nagasawa, Y.; Okada, T.; Sato, M.; Kohzuma, T. J. Chin. Chem. Soc. 2000, 47, 693; Chem. Phys. Lett. 2000, 331, 396. (32) (a) Scherer, P. O. J.; Fischer, S. F. Chem. Phys. 1989, 131, 115. (b) Scherer, P. O. J.; Scharnagl, C.; Fischer, S. F. Chem. Phys. 1995, 197, 333. (c) Scherer, P. O. J.; Fischer, S. F. Spectrochim. Acta, Part A 1998, 54, 1191. (33) Nakagawa, H.; Okada, T.; Koyama, Y. Biospectroscopy 1995, 1, 169. (34) Hasegawa, J.; Nakatsuji, H. J. Chem. Phys. B 1998, 102, 10420. (35) For most recent example, see: Chang, C. H.; Hayashi, M.; Liang, K. K.; Chang, R.; Lin, S. H. J. Phys. Chem. B 2001, 105, 1216, and references therein. (36) Michel-Beyerle, M. E.; et al. Private communication, 1996. See also Fig.51 in ref 3a as reproduction of one of their main results. (37) Huber, H.; Meyer, M.; Na¨gele, T.; Hartl, I.; Scheer, H.; Zinth, W. Chem. Phys. 1995, 197, 297. (38) (a) Allen, J. P.; Williams, J. C. J. Bioenerg. Biomembr. 1995, 27, 275. (b) Woodbury, N. W.; Lin, S.; Lin, X.; Peloquin, J. M.; Taguchi, A. K. W.; Williams, J. C.; Allen, J. P. Chem. Phys. 1995, 197, 405. (c) Mattiolli,

9622 J. Phys. Chem. B, Vol. 105, No. 39, 2001 T. A.; Lin, X.; Allen, J. P.; Williams, J. C. Biochemistry 1995, 34, 6142. (39) Artz, K.; Williams, J. C.; Allen, J. P.; Lendzian, F.; Rautter, J.; Lubitz, W. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 13582. (40) Photosynthetic organs are composed of an assembly of many units with the same structure such as antenna systems and a reaction center encircled by them. Energy structures of individual units are different little by little due to static structural disorder in each relevant protein-pigment complex.3,4,27c,d The present formula eq 5.30 gives the rate constant in each

Sumi and Kakitani single unit, but it varies little by little from a unit to a unit. In these situations, the time decay of the observed survival probability at the donor state should become multiexponential. The initial rate of the decay is given by the average of the rate constants of individual units. The first passage time is given by the average of the reaction times of individual units. The inverse of the first passage time gives the average rate constant. In this way, eq 5.30 enables us to obtain fundamental physical quantities of the system under investigation, in these cases.

Unified Theory on Rates for Electron Transfer Mediated by a Midway

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