J. Phys. Chem. 1996, 100, 12311-12320
12311
Superexchange Coupling and Electron Transfer in Homogeneous Three-Dimensional Systems: Site and Bond Disorders A. Filali-Mouhim, J.-M. Lopez-Castillo, I. L. Plante, and J.-P. Jay-Gerin* Groupe du Conseil de Recherches Me´ dicales du Canada en Sciences des Radiations et De´ partement de Me´ decine Nucle´ aire et de Radiobiologie, Faculte´ de Me´ decine, UniVersite´ de Sherbrooke, Sherbrooke, Que´ bec J1H 5N4, Canada ReceiVed: NoVember 28, 1995; In Final Form: March 3, 1996X
The superexchange coupling (VS) between two distant redox centers in large three-dimensional disordered media is studied by considering an electron-transfer model system made up of an aperiodic array of sites (s-type orbitals) interacting through an exponentially decaying function of distance. The associated oneparticle, tight-binding Hamiltonian accounts for the simultaneous presence of both diagonal and off-diagonal disorders, leading to a stochastic variation of VS. Exploring the statistical properties of VS under variations of the orbital energies and couplings, we determine the conditions for self-averaged behavior of VS and the amplitude of its fluctuations. For systems with wide electronic bands, VS is found to exhibit large fluctuations, while these latter are small for narrow-band systems. Moreover, for a given bandwidth of the intermediate medium, systems with a higher density of sites behave in a more self-averaged way. An analytical description of the average behavior of VS and of its fluctuations is provided in the narrow-band regime.
I. Introduction
k)
Electron transfers (ET) are fundamental steps of many biological processes, including respiration and photosynthesis.1 In many cases, they occur between distant oxidation-reduction (redox) centers embedded in a bridging protein medium and involve the tunneling of an electron over large distances (up to 20-25 Å, typically).1 Because of such large separations, the electronic coupling between the two redox centers, which is responsible for the tunneling, is generally very weak and it is expected to be strongly dependent on the nature of the intervening medium. Many experiments have been performed on ET reactions in biological molecules such as proteins1 to understand the role of this intervening medium. Two major answers have been given to this problem. The first one is based on a number of observations in photosynthetic complex systems and in semisynthetic ruthenium-cytochrome c and ruthenium-myoglobin systems2 where the bridging medium is found to behave as a mere uniform barrier through which the electron tunnels. In this case, the structural and electronic details of the coupling medium appear to be unimportant. The second one concerns certain molecules, such as ruthenium-modified cytochrome c, myoglobin, and cytochrome b53 or azurins,4 where the electron transfer occurs along certain specific pathways. In that case, the ET coupling appears to depend strongly on the structural and electronic details of the intervening medium. Although this latter electronic coupling pathway picture has received much attention from recent theoretical studies,3a,5 the former ET mechanism rests on a simple transposition of the Gamow tunneling model.6 It is not clear, yet, how complex molecular systems such as proteins could act as simple potential barriers while capturing the major effects of electronic control. It is the purpose of this paper to provide some insight into this problem. Formally, in the nonadiabatic limit, the ET reaction rate constant k is given by a generalized version of Fermi’s Golden Rule X
Abstract published in AdVance ACS Abstracts, June 1, 1996.
S0022-3654(95)03514-3 CCC: $12.00
2π |T |2(FC) p DA
(1)
where p is Planck’s constant divided by 2π, FC is the so-called Franck-Condon factor, a measure of the nuclear motion coupled to the ET process, and TDA is the electronic transition matrix element between the reductant (donor, D) and the oxidant (acceptor, A). The present work is devoted only to the study of this electronic term and, more precisely, to the influence of the intermediate medium on it. TDA is given by the sum of the direct through-space interaction VDA between the two redox sites and of the superexchange coupling VS that contains the information about the electronic properties (orbital energies and their couplings) of the intervening medium. The direct term VDA is assumed to be negligible for long-range electron transfers. The superexchange term VS is given by7
VS ) VATG(E0)VD
(2)
where G(E0) is the Green’s function8 of the medium, E0 is the energy of the donor and acceptor (that is, the acceptor and donor orbitals have the same energy), and VD and VA are the vectors of interaction of the donor and acceptor with the medium, respectively. The motivation of our work is to determine the general features of the electronic superexchange coupling VS in large three-dimensional (3-D) ET systems. For this purpose, we propose to derive such information from the systematic study of model systems that present interesting similarities with more complex real molecular systems. Among these similarities, we will retain the 3-D geometry and the aperiodic or “topologically” disordered structure of the system, the large number of medium sites involved during the electron transfer, and the diversity of chemical components. By including this chemical (or compositional) diversity, the present study is a continuation of a previous paper,9 hereafter referred to as I, where we studied in detail the case of a singlecomponent model system (all the medium orbitals having the same energy), using both numerical calculations and an analytical approach based on what we called the “continuous-medium © 1996 American Chemical Society
12312 J. Phys. Chem., Vol. 100, No. 30, 1996
Filali-Mouhim et al. electron or a hole, so that we exclude from this study the investigation of many-body effects. A probability distribution is chosen for the random variables Ei (orbital energies) and Vij (coupling prefactors). This choice determines the nature of the bridging medium. We also assume that the donor and acceptor sites are coupled each to a single site on the surface of the sphere that we naturally call the “initial” and “final” sites, noted I and F, respectively (see Figure 1), so that VS can be expressed as
VS ) VAVDGIF(E0)
Figure 1. Projection of the studied model electron-transfer system on the z ) 0 plane. The donor site (noted D) is coupled to the initial or “entrance” site (noted I) through a bond of strength VD. The acceptor site (noted A) is coupled to the final or “exit” site (noted F) through a bond of strength VA. The electron-transfer distance d is given by the distance between the I and F sites. The number of orbital sites in the system is N ) 932. Note that this structure was generated by a molecular dynamics calculation carried out with an ensemble of 2048 argon atoms interacting in pairs according to the Lennard-Jones 12-6 potential with length and energy parameters given by σ ) 3.405 Å and /kB ) 119.8 K, respectively (see text and ref 11).
approximation” (CMA). In particular, the CMA allowed us to derive an expression for the decay coefficient (βeff) of VS as a function of the ET distance (d) (the success of this approximation was demonstrated by a comparison to the recent ET calculations of Evenson and Karplus10 on a simplified model of tuna cytochrome c). In the present paper, we now generalize our study to include the disorder introduced by homogeneous chemical substitutions, but still disregarding the role of the spatial structure of the intervening medium. The paper is organized as follows. In section II, we describe the model system and the principles of our statistical analysis. Section III contains the theoretical development of this study. The numerical results are presented in section IV, along with the tests of the approximations introduced in section III. Finally, our conclusions are given in section V.
(5)
VS may be viewed as a random variable which is related to the particular values given to Ei and Vij. The important questions that arise are since, for a given probability distribution of the random variables Ei and Vij, many spatial configurations of the system can be obtained, how much and in what way will VS depend on a particular configuration? The relation of these questions to the problem of superexchange coupling in disordered macromolecules, such as proteins, may be formulated as follows. Proteins represent very large molecular structures that are built up from the same basic constituents (20 different amino acids) and that differ, at least partly, in the number, sequence, and positions of these. Even a small ET protein such as cytochrome c is composed of about 100 amino acids. When constructing the electronic Hamiltonian of the protein medium, various strategies may be used,13-15 leading to a huge matrix whose elements are subjected to a high degree of imprecision. One may then ask how will this imprecision be reflected on the electronic coupling value? Moreover, interchanging the relative positions of certain peptide components will result in matrices still sharing important similarities. The question here is how will VS be affected by such permutations? We attempt to address these questions below. III. Theory The Hamiltonian of the medium system can be decomposed as follows:
H ) H0 + V
(6)
II. Model System Our ET model system (see Figure 1) consists of an aperiodic array of N sites (s-type orbitals) of energy Ei located inside a sphere of diameter d. The site coordinates are generated by means of a standard molecular dynamics simulation routine for a Lennard-Jones fluid.11 The sites interact with one another through an exponential coupling law
V(b r i, b r j) ) Vije-β|br i-br j|
(3)
where β is the decay coefficient of the interaction (throughout ri and b rj are this paper, β-1 will be used as the unit of length), b the coordinates of the interacting sites i and j, respectively, and Vij is a coupling prefactor. Assuming that all sites are identical (that is, Ei is a constant, and so is Vij) would recover the ET model system thoroughly studied in I. One interesting feature of this exponential interaction law is its widespread use in the description of site-site interactions in proteins.10,12,13 The present model system is described by the tight-binding Hamiltonian N
r i,b r j)|i〉〈j| H ) ∑Ei|i〉〈i| + ∑V(b i)1
(4)
i*j
H is an effective one-particle Hamiltonian, describing either an
where H0 is the “diagonal” free Hamiltonian
(H0)ij ) Eiδij
(7)
and V is the “off-diagonal” intersite interaction. Equation 5 can be rewritten as
VS ) VAVD〈I|(E0 - H)-1|F〉
(8)
If E0 is distinct from all eigenvalues of H, the Green’s function for the medium can be expressed in terms of the Born expansion
G ) G0 + G0VG0 + G0VG0VG0 + G0VG0VG0VG0 + ... (9) where G0 is the free Green’s function defined by
(G0)ij )
δij E0 - Ei
(10)
From now on, we set E0 ) 0 as the reference energy to simplify the notations. Ei will then denote the energy separation between the orbital on site i in the bridge and the orbitals of the donor and acceptor sites. Inserting the closure relation between each pair of operators in eq 9, we get
Superexchange Coupling and Electron Transfer
J. Phys. Chem., Vol. 100, No. 30, 1996 12313
〈∏ 〉 〈 〉
V(b r I,b r F) V(b r I,b r i)V(b r i, b r F) -1 GIF ) δIF + -∑ + EI EIEF EIEiEF i V(b r I,b r i)V(b r i, b r j)V(b r j,b r F)
∑ij
N(C)+1
i)1
+ ... (11)
EIEiEjEF
where each term of the series defines a path (that is, a sequence of sites) starting at site I and ending at site F. Substituting the summation over sites i, j, k, ... by the formal summation over paths and exploiting the exponential form of the coupling given in eq 3, VS can be rewritten as N(C) N(C)+1 -βL(C)
VS ) VAVD∑(-1)
N(C)+1
1
Vii+1 ∏ ∏ i)1 i)1 E
e
C
(12)
i
where N(C) is the number of jumps in path C, and L(C) ) ∑rii+1 is the path length. In eq 12, Vii+1 denotes the coupling prefactor between two consecutive sites along the path, omitting, for the sake of simplicity, reference to the path. A more correct notation would have replaced the subscript i by i(C). In large 3-D ET systems, it is expected that a correspondingly large number of paths will contribute to the summation in eq 12, thus possibly leading to some kind of smearing of the particular values of the bridge orbital energies and couplings. If this were the case, VS would not differ strongly from one spatial configuration of the system to another, and a good approximation to VS would be obtained by averaging over all possible configurations. Under certain conditions, one could observe that VS is a “self-averaged” quantity much like the electronic density of states in disordered media, to mention only one important example.16 These considerations constitute the starting point of the present study where we determine the conditions under which VS may present a self-averaged behavior. In the following, our approach then consists to first characterize the expectation value of VS and to turn next to the evaluation of its fluctuations. A. The Average Medium. Two kinds of averaging are considered, one over the chemical or compositional (diagonal and off-diagonal) disorder and the other over the spatial or topological disorder. The former will be calculated using the “virtual-medium approximation” (VMA), and the latter using the CMA.7 This averaging is expressed by the following notation:
VS S 〈〈VS〉(Ei,Vij)〉(b) r
(13)
1. The Virtual-Medium Approximation. Averaging first eq 12 over the distributions of energies and couplings, with the assumption that these quantities are independent stochastic variables (correlation effects will be studied below), we get
〈
VS ) VAVD
∑C e
N(C)
-βL(C)
〈∏ Vii+1〉
N(C)+1
(-1)
i)1
〈 〉〉 N(C)+1
i
(14) (b) r
Assuming next that the most important set of paths in the ET process is that of self-avoiding paths,9 we have the following relations N(C)
〈∏ Vii+1〉 ) 〈Vij〉N(C) i)1
Ei
)
1
N(C)+1
(15)
(16)
Ei
each expectation value being independent of the particular bond or site considered, respectively. Now, making the approximation that similar relations hold for all paths C, we obtain
〈
VS ) VAVD
〉
VN(C) -βL(C) N(C)+1 e (-1) ∑C EN(C)+1
(17) (b) r
where we have introduced the average coupling prefactor V ) 〈Vij〉 and the average inverse energy 1/E ) 〈1/Ei〉. Equation 17 is the configurational average of VS for an effective intermediate medium characterized by a single orbital energy separation E and a single intersite coupling prefactor V. This approximation is very similar to the so-called “virtual-crystal approximation” for the Green’s function of disordered systems, such as crystal alloys or mixed crystals.8,16 We do not not use this terminology here, however, because our calculations do not make any reference to a crystal structure. Rather, we will adopt the related expression “virtual-medium approximation” (VMA). We should note that the present derivation of the VMA differs from the usual one where one defines an average energy given by 〈Ei〉 instead of an average inverse energy. The usual procedure intends to describe cases of small disorder in an otherwise unperturbed electronic band.8,16 Our approach is correct in the strong disorder limit as long as self-avoiding paths dominate the ET process. This is expected to be true for large energy separations, that is, |E| . |V|. An important limitation of the VMA should be mentioned here. It concerns the question of the sign of the coupling prefactors. This sign reflects the nodal structure and orientation of the orbitals,17 which has no effect in the present study given that we are dealing with s-type orbitals only. It is clear that |VS| should be invariant with respect to the sign convention or, in other words, to the phase of the orbitals. Clearly, this is not true concerning the VMA, unless all Vij share the same sign. Although this is an important restriction for the applicability of our treatment, it must be noted that current ET calculations in proteins use the same assumption.10,13 2. The Continuous-Medium Approximation. Equation 17 has been studied in detail in I. The configurational average of VS has been evaluated by assuming an infinite, structureless medium. Briefly, the CMA amounts to replacing the discrete Hamiltonian of eq 4 by its continuous form r b′| r r |b〉〈 r b| r + F2∫db r db′ r Ve-β|b|b〉〈 r b′| r (18) H ) FE∫db
where F is the density of orbital sites and the integration is over the entire space. Introducing the width of the continuousmedium electronic band given by
|Γ| ) 8πF|V|β-3
1
∏ i)1 E
1
(19)
we get the following expressions, according to the sign of E/Γ:9
(i) for E/Γ > 0: VCMA )
βd E Γ 1 + - 1) × sin( x x x Γ E E βd x2 βd Γ 1 + + 1) (20) exp(x x E x2
2VAVDV 2
12314 J. Phys. Chem., Vol. 100, No. 30, 1996
Filali-Mouhim et al.
(ii) for E/Γ < 0: VCMA )
E Γ exp(-βd 1 { | | | |) x x x Γ E E βd exp(-βd 1 + x x|EΓ|)} (21)
VAVDV 2
Since we are dealing with s-type orbitals, the intersite couplings are negative8,17 (note that in proteins it is also usual to take all Vij < 010,13). Under this condition, Γ < 0, and eqs 20 and 21 correspond to hole (E < 0)- and electron (E > 0)mediated transfers, respectively. Of course, the terminology should be inverted for Vij > 0. As is clearly seen, the sign of E/Γ determines two drastically different behaviors of the superexchange coupling. From eqs 20 and 21, we can observe that the ratio VS/Vdir, where
Vdir )
VAVDV -βd e E2
(22)
is the smallest path (consisting of a single jump between the initial and final sites) contribution to VS, is a function, for a given distance, of the single parameter E/Γ. This dimensionless “control parameter” for 3-D electron-transfer systems is similar to the corresponding one previously found for one-dimensional systems.18 This result shows the universal character of this parameter in controlling the superexchange coupling. Although eqs 20 and 21 have been discussed in detail in I, it is useful to recall here their main features. Figure 2 shows, in this regard, a typical example of the variation of VS/Vdir as a function of E/Γ. We see that, due to interferences between many paths, the behavior of VS/Vdir for the case E/Γ < 0 (electron transfer) is drastically different from that for the case E/Γ > 0 (hole transfer). In fact, the effective electronic coupling differs by many orders of magnitude and, while presenting a monotonic variation in the first case, shows an oscillatory behavior in the second. Equations 20 and 21 describe the configurational average of VS. However, each configuration may deviate from the average behavior and we thus now turn to the evaluation of the importance of fluctuations. B. Fluctuations. The evaluation of the fluctuations of VS may be carried out by following the same lines of reasoning as in section III.A. We start with the formal definition
δ2VS ) 〈VS2〉 - 〈VS〉2
(23)
where the brackets denote the average over the energy and coupling distributions. From eq 12, we may write
〈VS2〉 ) (VAVD)2 ∑ e-β[L(C)+L(C ′)](-1)N(C)+N(C ′) × C,C ′
〈∏
N(C)
i)1
N(C ′)
N(C)+1
Vii+1 ∏ Vjj+1 j)1
1
N(C ′)+1
∏ ∏ i)1 E j)1 i
1
Ej
〉
Figure 2. Typical variation of |VS/Vdir| as a function of the control parameter E/Γ in the case of a one-component system, for hole and electron transfers. (9) Exact numerical results; (s) CMA predictions. The orbital energy |E| varies between 2.5 and 25, the donor-acceptor distance is βd ) 15, the density of sites is Fβ-3 ) 0.1, and V ) -1.
VN(C)+N(C ′)-2p δ2VS ) (VAVD)2 ∑ e-β[L(C)+L(C ′)] C,C ′ EN(C)+N(C ′)-2q+2 p(C,C ′) q(C,C ′) 1 V2p 2 Vii+1 ∏ (25) ∏ 2 i)1 i)1 E E2q i
〈
where p(C,C ′) and q(C,C ′) are the number of common links and sites of paths C and C ′, respectively. δ2VS can be rewritten as
δ2VS ) ∑δ2Vp,q
Applying the VMA, that is, neglecting higher-order correlations in eq 24, we get
(26)
p,q
where we have introduced the quantity δ2Vp,q. A comparison of eqs 25 and 26 shows that δ2Vp,q represents the contribution of all pairs of paths C, C ′ with fixed p and q values. We now assume that the most important contribution to eq 25 comes from the lowest order terms with minimum values of p and q. This, as for the VMA and the CMA, is a narrow-band approximation. Because of our system’s configuration, all paths share in common the sites I and F, so that q g 2, and EI and EF systematically contribute in all terms of eq 25. These terminal sites will play an important role and, in this sense, possess an intrinsic specificity. It is clear, moreover, that their couplings to the rest of the medium will also enter eq 25 in a systematic manner, thus reinforcing their specific role. In the following, we will be interested in the contribution of the bulk medium, so that it is desirable to get rid of the fluctuations introduced by the I and F sites. This is simply done by setting EI ) EF ) E ()constant) and VIi ) VjF ) VIF ) V ()constant). 1. Diagonal Disorder. The lowest order contribution from diagonal disorder (that is, fluctuations of VS due to sites’ energy fluctuations) comes from single crossings of paths inside the intervening medium between I and F (p ) 2, q ) 3). The summation over C in eq 25 then reduces to a summation over all possible crossing sites (M), and we get
δ2V2,3 ) (VAVD)2 (24)
〉
( ) ( )∑ V
E
4
δ2
1
Ei
e-2βrIMe-2βrMF
(27)
M
where δ2(1/Ei) is the variance of 1/Ei. This expression is implicitly related to the actual spatial structure of the intervening medium, so that we proceed by applying the CMA. The discrete summation in eq 27 can then
Superexchange Coupling and Electron Transfer
J. Phys. Chem., Vol. 100, No. 30, 1996 12315
be replaced by its continuous approximation
δ2V2,3 ) F(VAVD)2
( ) ( )∫ V4 2 1 δ E Ei
r -2β|b-d r B| db r e-2β|b| e
(28)
()
(29)
which is readily integrated to give
()
δVS F )V 3 VS β
1/2
[f(βd)]1/2δ
1 Ei
where f is a function of βd only, defined as
f(βd) )
4 π 1 + 2βd + (βd)2 8 3
[
]
(30)
In the particular case of a binary mixture of sites with concentrations x1 and x2, the orbital energies will assume two values E1 and E2 ) E1 + ∆E with probabilities x1 and x2 ) 1 - x1, respectively, so that δVS/VS takes the following form:
()
δVS F ) xx1x2 3 VS β
1/2
[f(βd)]1/2
| | V∆E E1E2
(31)
δ V3,4 ) (VAVD)
2
( ) ( )∑ V
E
4
δ
2
Vij
EiEj
e
-2βrIM -2βrMN -2βrNF
e
e
(32)
M,N
where δ2(Vij/EiEj), the variance of Vij/EiEj, combines diagonal and off-diagonal disorders. Again, applying the CMA, we get
δ2V3,4 )
( ) ( )∫
F2(VAVD)2
V 4 2 Vij δ E EiEj
r -2β|br b′| r -2β|b′-d r B| db r db′ r e-2β|b| e e
(33) leading to
()
( )
Vij δVS F ) V 3 [g(βd)]1/2δ VS EiEj β
(34)
where g is a function of βd only, given by
g(βd) )
12 16 16 7π2 1 + 2βd + (βd)2 + (βd)3 + (βd)4 128 7 21 105 (35)
[
]
In the case of a system with a single orbital energy E (that is, with no diagonal disorder) and a binary distribution of coupling prefactors assuming the values V1 and V2 ) V1 + ∆V, with probabilities y1 and y2 ) 1 - y1, respectively, δVS/VS takes the following form:
( )| |
δVS F V∆V ) xy1y2[g(βd)]1/2 3 VS β E2
fσ(βd) ) r -2β|b-d r B| r Θ(r-σ) Θ(|b r -B d | - σ)e-2β|b| e (37) β3e2βd∫db
r db′ r Θ(r-σ) Θ(|br b′|-σ) r × gσ(βd) ) β6e2βd∫db r -2β|br b′| r -2β|b′-d r B| Θ(|b′-d r B|-σ)e-2β|b| e e (38)
Of special importance, we note from eq 31 a characteristic square-root dependence of δVS/VS on the concentration of each energy component. 2. Off-Diagonal Disorder. The lowest order contribution from off-diagonal disorder (that is, fluctuations of VS due to coupling prefactor fluctuations) arises from paths sharing a single common link inside the intervening medium between I and F (p ) 3, q ) 4). The summation over C in eq 25 then reduces to a summation over all possible pairs of sites (M and N), and we get 2
where, again, we observe a characteristic square-root dependence of δVS/VS on the concentration of each coupling component. We should also note from eq 34 the role played on δVS/VS by the density of orbital sites, which is in contrast to that observed in eq 29. 3. Structural Effects. Equations 29 and 34 do not take into account the spatial structure of the intermediate medium. This situation may be partly remedied by introducing the pair distribution function g(r) of the system, which thus amounts to define another effective continuous medium. Using, for simplicity, the zero-density limit of the pair distribution function of a hard-sphere fluid model g(r) ) Θ(r-σ), where Θ is the step function and σ is the diameter of the volume (assumed to be a sphere) occupied by a site (that is, 1/F ) πσ3/6), we obtain two other functions, fσ and gσ, given by
(36)
which generalize eqs 30 and 35, respectively. Even introduction of such a simple pair distribution function does not allow an analytical evaluation of eqs 37 and 38. Instead, these equations can easily be resolved numerically. The effect of the pair distribution function on δVS/VS will be discussed in the next section. It is worth noting here that the CMA corresponds to the case σ ) 0. IV. Numerical Results Because of the great number of parameters describing the general ET system, it is necessary to limit our numerical studies to a few representative and interesting situations. From this viewpoint, we thus focus our attention on the following cases: (1) For a given donor-acceptor distance and a given density of sites, the average behavior and the fluctuations of VS are studied in the case of a binary ET system with both diagonal and off-diagonal disorders. The validity of the VMA is thus tested. A compilation of numerous calculations at different densities is given. (2) The effect of correlations between orbital energies and coupling prefactors is investigated in the case of a twocomponent ET system. (3) The variation of VS and δVS/VS as a function of the donor-acceptor distance is illustrated in the case of a 10component ET system. A. Site and Bond Disorders: The Two-Component System. To proceed with the exact numerical calculations, we study a model system with binary distributions of orbital energies and coupling prefactors. Thus, Ei assumes the values E1 and E2 with probabilities x1 and x2 ) (1 - x1), respectively, and similarly Vij assumes the values V1 and V2 with probabilities y1 and y2 ) (1 - y1), respectively. To concentrate on the bulk properties of the intermediate medium, the fluctuations introduced by the I and F terminal sites are avoided by fixing EI ) EF ) E1, VIi ) VjF ) V ) yV1 + (1 - y)V2. In what follows, the donor-acceptor distance and the density of sites are given by βd ) 15 and Fβ-3 ) 0.1, so that the number of orbitals is N ) 176. As mentioned in section III, it is most convenient to study the ratio VS/Vdir. 1. Test of the VMA. Figure 3 presents a comparison of VS/Vdir as obtained from the average over 100 orbital energy configurations and from the VMA. Results are shown as a
12316 J. Phys. Chem., Vol. 100, No. 30, 1996
Figure 3. Illustration of the validity and limits of the virtual-medium approximation in the case of a two-component system. Plot of |VS/Vdir| as a function of the control parameter E/Γ, for both hole- and electron-mediated transfer. (2) Exact numerical results; (3) VMA predictions. |E1| varies from 5 × 10-3 to 5 × 102, |E2| ) |E1| + 5, the coupling prefactors are V1 ) -1 and V2 ) -5, the concentration parameters are x1 ) y1 ) 0.5, the donor-acceptor distance is βd ) 15, and the density of sites is Fβ-3 ) 0.1. The average is over 100 spatial site configurations.
function of the control parameter E/Γ for both electron- and hole-mediated transfers. The VMA is seen to be in excellent agreement with the exact numerical results. This might be somewhat surprising in view of the drastic assumption that characterizes the VMA, namely, the predominance of selfavoiding paths in the ET process. But one may rationalize as follows. In the narrow-band limit (|E/Γ| . 1), one expects that the repeated passage of an electron over a same site, or along a same bond, is a rather costly operation so that only selfavoided paths actually contribute to the transfer. In contrast, in the wide-band limit (|E/Γ| ∼ 1), an electron not only has the ease to visit repeatedly the same site or the same bond but can also easily explore neighboring and remote uncorrelated sites and bonds of the intervening medium. The existence of a great number of different sites (we should remember that we are considering large 3-D ET systems) appears to be determinant in that respect. The VMA seems to break down only when approaching a resonance condition, that is, when some energy level of the intervening medium is close to that of the donor and acceptor. It should be noted, in particular, that the VMA fails to describe the resonance threshold. This is to be expected, since it is wellknown from the theory of disordered systems that the virtualcrystal approximation affords a rather poor description of the bandwidth.16 2. Fluctuations. Although the VMA correctly describes the average behavior of VS, any actual configuration may differ from this average. Thus, attention must be paid to such deviations from the mean. The importance of fluctuations will be best monitored by looking at the ratio δVS/|VS|. As can be seen from Figure 3, VS may vanish in the case of hole transfer (E/Γ > 0), so that we will limit our presentation to the case of electron transfer (E/Γ < 0). Note, however, that outside a limited region around that of vanishing VS, the relative fluctuations δVS/|VS| are well defined in both cases and are found to behave in a similar way. Let us first consider the case of diagonal disorder. Figure 4 shows δVS/|VS| as a function of x1 at two values of the density. The choice of parameters is such that |E/Γ| ranges from 1 to 10 as x1 varies from 0 to 1, thus covering the wide- to narrowband domains. We note that fluctuations are smaller at higher density. This is an interesting behavior since large ET systems,
Filali-Mouhim et al.
Figure 4. Typical variation of the relative standard deviation δVS/|VS| as a function of the concentration parameter x1 for two values of the density of sites, in the case of a two-component system with diagonal disorder only. (4) Fβ-3 ) 0.1, E1 ) 2.5, E2 ) 25; (2) Fβ-3 ) 0.5, E1 ) 12, E2 ) 125. (+) and (s) are calculated results from eq 29 for Fβ-3 ) 0.1 and 0.5, respectively. The donor-acceptor distance is βd ) 15 and the coupling prefactor is V ) -1. The average is over 100 spatial site configurations.
TABLE 1: Diagonal Disorder. Illustration of the Structural Effects on the Fluctuations of VS for Two Values of the Density of Sitesa Fβ-3 ) 0.1 R0 Rσ
Fβ-3 ) 0.5
E/Γ ) 1.1
E/Γ ) 8.42
E/Γ ) 1.1
E/Γ ) 8.42
2.43 2.25
1.36 1.25
1.48 1.45
1.14 1.12
a The ratio R ) (δV /|V |) /(δV /|V |) σ S S σ S S num is calculated as described in the text with the same data as for Figure 4. Note that R0 corresponds to σ ) 0 and the values of βσ for the calculation of Rσ are 2.67 and 1.56 for Fβ-3 ) 0.1 and 0.5, respectively.
such as proteins, with a high density of orbital sites will present a higher degree of self-averaging and are best described by mean-field approximations such as the CMA. Our theoretical prediction of eq 31 is also shown in Figure 4. Although the overall variation and the order of magnitude are correctly described, there are some discrepancies originating from two sources. One is the narrow-band approximation used in the derivation of eq 31. The agreement is slightly better near x1 ) 1 (|E/Γ| ) 10) than near x1 ) 0 (|E/Γ| ) 1). The other is structural and is dominated by the correlation hole of the hard-sphere fluid model of the pair distribution function. This latter effect is confirmed in Table 1 where the ratio Rσ of our theoretical estimation of (δVS/|VS|)σ (eqs 31 and 37) to the exact numerical fluctuations observed (δVS/|VS|)num is evaluated at vanishing and finite σ values. It is seen that Rσ is closer to 1 when the correlation hole effect is taken into account (finite σ value). Again, we note that structural effects become less important as the density of the system is increased. We have also considered the case of off-diagonal disorder (eqs 36 and 38), obtaining essentially the same results and conclusions. We will thus not discuss this case further. We should only mention here, however, that structural effects are more important for off-diagonal disorder (see Table 2) than for diagonal disorder. This readily transpires from a comparison of eqs 37 and 38 where it is seen that the pair distribution function enters two and three times in the structural factors fσ and gσ, respectively. 3. The Law of Corresponding States. Figures 5 and 6 show compilations of a number of calculations of the mean value VS/Vdir as a function of E/Γ, concerning an ensemble of binary ET systems with a degree of disorder more important than those studied above and involving simultaneous diagonal and offdiagonal disorders. Taking V1 ) -1 and βd ) 15, the free variables involved in the problem are E1, E2, V2, x1, y1, and F.
Superexchange Coupling and Electron Transfer
J. Phys. Chem., Vol. 100, No. 30, 1996 12317
TABLE 2: Off-Diagonal Disorder. Illustration of the Structural Effects on the Fluctuations of VS for Two Values of the Density of Sitesa Fβ-3 ) 0.1 R0 Rσ
Fβ-3 ) 0.5
E/Γ ) 1.1
E/Γ ) 8.42
E/Γ ) 1.1
E/Γ ) 8.42
8.05 5.55
2.15 1.48
5.14 4.55
1.76 1.55
a The ratio Rσ ) (δVS/|VS|)σ/(δVS/|VS|)num is calculated as described in the text. Data used in the calculation are y1 ) 0.1 and y2 ) 0.9 for E/Γ ) 1.1, and y1 ) 0.98 and y2 ) 0.02 for E/Γ ) 8.42; V1 ) -1 and V2 ) -10. The donor-acceptor distance is given by βd ) 15 and the average is over 100 spatial configurations of the sites. Note that R0 corresponds to σ ) 0 and the values of βσ for the calculation of Rσ are 2.67 and 1.56 for Fβ-3 ) 0.1 and 0.5, respectively.
Figure 6. A measure of the self-averaged behavior of VS through the variation of the relative standard deviation δVS/|VS| as a function of the control parameter E/Γ. Parameters and symbols are as in Figure 5. The solid lines represent the theoretical predictions obtained from the combination of eqs 26, 30, 31, 35, and 36 for the binary distributions for Ei and Vij corresponding to the concentration parameters x1 ) y1 ) 0.5. These values approximately give the maximum standard deviation for a given value of E/Γ.
Figure 5. Evidence for a close correlation between |VS/Vdir| and the control parameter |E/Γ| for hole and electron transfers. Each point in the figure is obtained in the following way: (i) a density of sites is selected (three values of the density Fβ-3, namely, 0.05 (2), 0.2 (0), and 0.5 (b), corresponding to 88, 353, and 883 sites in the sphere, respectively, are presented); (ii) a set of values of the parameters characterizing the probability distributions of the orbital energies and coupling prefactors is chosen in the following intervals: |E1| ∈ [1, 90], |E2| ∈ [|E1|, |E1| + 8], V2 ∈ [-5, -1], x1 ∈ [0, 1], and y1 ∈ [0, 1]; (iii) once these parameters are given, 10 independent configurations are generated by randomly sampling the energy of each site (E1 or E2) and the coupling prefactor of each pair of sites (V1 or V2); (iv) each such configuration yields a matrix H, which is numerically inverted to give VS; and (v) the average over these values is taken. The donor-acceptor distance is βd ) 15 and V1 ) -1. The solid line is the CMA prediction.
The different sets of parameters are randomly generated within a broad range of variation (see caption of Figure 5). Figure 5 confirms that, for a given donor-acceptor distance, a unique dimensionless parameter (that is, E/Γ) controls the average behavior of VS, as all the results are found to lie approximately on the same curve. In the narrow-band limit, this “law of corresponding states”19 is well observed. For wideband systems, it is seen that the density also influences the behavior of VS. The observed spreading is certainly due to a poor evaluation of the bandwidth which we have assumed to be given by the CMA. It would be interesting to include structural effects for a better determination of Γ, perhaps along the lines presented in I, but such a calculation remains very difficult. Figure 6 shows the variation of δVS/|VS| as a function of E/Γ. We observe that the electron and hole cases are very similar,
except naturally in the regions where VS vanishes (see Figure 5). Large fluctuations are seen to be associated with wide bands and small fluctuations with narrow bands. Moreover, systems of higher densities behave in a more self-averaged way, which is in the spirit of the present statistical study of large ET systems. Since the disordered binary medium behaves similarly to the effective one-component medium, the conclusions drawn in I for the latter apply as well to the former. In particular, we inferred in our preceding paper the existence of a “law of corresponding states” stating the equivalence of electron-transfer systems with different orbital energies, electronic couplings, and densities of sites. These systems are unified by the introduction of a unique control parameter E/Γ. This law of corresponding states is extended here to include ET systems presenting both diagonal and off-diagonal disorders as well. B. Correlation Effects. We now investigate the effect of introducing some degree of correlation between orbital energies and coupling prefactors on VS and δVS/|VS|. We assume that the couplings are entirely determined by the energies of the interacting sites through the well-known Wolfsberg-Helmholz formula20
(Ei + Ej) -β|br ij| V(b r i, b r j) ) R e 2
(39)
largely used in protein modeling studies,14 where R is a dimensionless coupling parameter. Using the notation of the preceding sections, we thus set
1 Vij ) R(Ei + Ej) 2
(40)
We limit our study to a two-component system, the corresponding orbital energies E1 and E2 being distributed with probabilities x1 and x2 ) (1 - x1), respectively. It is worth noting that, in this case, only one concentration parameter is needed to define the ET system. According to eq 40, the coupling prefactors
12318 J. Phys. Chem., Vol. 100, No. 30, 1996
Figure 7. Illustration of the effect of correlations between orbital energies and coupling prefactors on |VS/Vdir| as a function of the control parameter E/Γ in the case of a two-component system. (9) Exact numerical calculations; (0) VMA predictions. (a, c) Uncorrelated case and (b, d) Correlated case. The dimensionless coupling parameter R (see eq 39) varies between 1/33 and 1/3, and x1 ) 0.5. The orbital energies are E1 ) 5 and E2 ) 15, the donor-acceptor distance is βd ) 15, and the density of sites is Fβ-3 ) 0.1. The average is over 100 spatial site configurations.
assume three values V11, V22, and V12, with probabilities x12, (1 - x1)2, and 2x1(1 - x1), respectively. In the calculations, the concentration, the density of sites, and the donor-acceptor distance are kept fixed, so that the variation of E/Γ is governed by R, which is varied continuously. Our interest resides in comparing the correlated and uncorrelated cases, keeping the same probabilities for the orbital energies and coupling prefactors. In Figure 7, parts a and c and b and d illustrate the behavior of the superexchange coupling VS/Vdir as a function of E/Γ, for the uncorrelated and correlated cases, respectively. The VMA is seen to work very well for the uncorrelated case, as was already pointed out. However, the correlated case, as is to be expected, presents a deviation from the VMA prediction. This deviation is very small in the narrow-band limit but is marked in the wide-band region. Such a result indicates, once more, that VS is more sensitive to the details of the intervening medium in the wide-band region. In essence, the observed deviation comes from the inability of the VMA to correctly reproduce the threshold of resonance. This, again, is typical of the VMA, which is unable to accurately predict the width of the electronic band of disordered systems.16 Looking now at the fluctuations of VS, Figure 8 shows the variation of δVS/|VS| as a function of the concentration x1 for a fixed value of R. As we can see, the fluctuations are more important in the uncorrelated system. Interestingly, we note that this result constitutes a common feature of disordered media (see, for example, the disorder-induced line widths of Raman and infrared absorption spectra in mixed crystals21). C. Effect of the Terminal Sites [D-I] and [F-A]. We should first recall here the obvious role in the electronic superexchange coupling VS of the [D-I] and [F-A] couplings of the donor and acceptor sites to the “initial” and “final” sites of the intervening medium (see eq 5). The role of the terminal sites I and F is, however, less easily seen. As shown in section III.B, the energies of the sites I and F and their couplings to the rest of the medium contribute systematically to the fluctuations of VS. To investigate the effect introduced by the sites I and F, we study a binary ET system in the case where EI, EF, VIi, and VjF are taken as random variables as the rest of the matrix elements of H. Using the same parameters as those of Figure 8, we found that δVS/|VS| is 5 times greater than its
Filali-Mouhim et al.
Figure 8. Illustration of the effect of correlations between orbital energies and coupling prefactors on the relative standard deviation δVS/|VS| as a function of the concentration parameter x1 in the case of a two-component system. (9) Uncorrelated system; (0) correlated system. The coupling parameter R (see eq 39) is fixed at 0.1. The orbital energies are E1 ) 5 and E2 ) 15, the donor-acceptor distance is βd ) 15, and the density of sites is Fβ-3 ) 0.1. The average is over 100 spatial site configurations.
Figure 9. Variation of |VS/Vdir| as a function of the donor-acceptor distance (βd) in the case of a 10-component system, for hole and electron transfers, and for two values of the control parameter |E/Γ| ) 2.46 (Fβ-3 ) 0.2) and |E/Γ| ) 9.85 (Fβ-3 ) 0.05). (2) Exact numerical results; (3) VMA results; (s) CMA results. The orbital energies are such that |Ei| ) 10 + 5(i - 1) with i ∈ [1, 10], and the concentration parameters are xi ) 0.1. The coupling prefactors are uniformly distributed in the interval [1, 3]. The average is over 100 spatial site configurations.
corresponding value in the case where EI, EF, VIi, and VjF are kept fixed. Clearly, this is a manifestation of the specific role played by the terminal groups [D-I] and [F-A]. We should note, however, that in real ET systems the donor and acceptor sites are usually linked to many “terminal” sites of the intermediate medium and not solely to a single “initial” or “final” site. In this latter case, we may readily expect that any given “terminal” site will play a less specific role. D. Distance Dependence. The donor-acceptor distance dependence of VS has hitherto been the subject of a number of experimental studies, which have recently led, in the case of biological ET systems and model proteins, to a controversy concerning the influence of the intervening medium on VS.2,3a,5b,10 We investigate this problem here in the case of a 10component ET system with both site and bond disorders. Figure 9 shows VS/Vdir as a function of the donor-acceptor distance for two values of the control parameter E/Γ. For both electronand hole-mediated transfers, we observe a globally exponential dependence of the effective coupling, which is to be expected for homogeneous systems. However, for the case E/Γ > 0 (hole
Superexchange Coupling and Electron Transfer
Figure 10. Relative standard deviation δVS/|VS| as a function of the donor-acceptor distance (βd), in the case of a 10-component system and for electron transfer. (b) |E/Γ| ) 2.46; (9) |E/Γ| ) 9.85. The orbital energies are such that |Ei| ) 10 + 5(i - 1) with i ∈ [1, 10], and the concentration parameters are xi ) 0.1. The coupling prefactors are uniformly distributed in the interval [1, 3]. The average is over 100 spatial site configurations.
transfer), this exponential behavior is altered by the presence of an oscillatory structure (see I and section III.A). As can be seen from the figure, the VMA works well at all distances. The prediction of the CMA, also shown in Figure 9, behaves remarkably well, in particular in the narrow-band limit. For the wide-band case, even though the CMA does not closely agree with the exact numerical results, the decay coefficient of the superexchange coupling, given by the slope of the curves, is reproduced in a quantitative way. This agreement is similar to the one reported in the case of a more complex structure such as that of the simplified model of tuna cytochrome c studied by Evenson and Karplus10 (see I and ref 22). Turning to the study of fluctuations, Figure 10 shows δVS/|VS| as a function of distance. As we can see, for a given value of E/Γ, δVS/|VS| increases weakly with distance. This reflects the fact that increasing the donor-acceptor distance increases the number of orbital sites that contribute to the fluctuations of VS (see eqs 27 and 32). Moreover, for a given donor-acceptor distance, the fluctuations increase when passing from the wide-band regime to the narrow-band region, as predicted by our calculations presented in section III.B. V. Conclusions It is generally admitted that proteins and most biological macromolecules present a high degree of specificity related to their function and structure. This property, if it were true for all physical and chemical properties, would greatly limit the suitability of a theoretical analysis. Especially concerning the electronic coupling between two remote redox centers embedded in a biological medium, a great number of microscopic parameters would need to be known with a high degree of accuracy to understand or predict the behavior of the system. These would include the precise knowledge of the nuclei’s positions and the detailed distributions of the bridge orbital energies and couplings, including many-body interactions, etc. In the present work, we have analyzed the relevance of looking for a perfect knowledge of the intervening medium’s microscopic parameters for the determination of the electronic coupling. We have shown that, under the conditions of the socalled “narrow-band” regime, the intervening medium behaves almost in a self-averaged way. In this regime, positions, orbital energies, and couplings need not be known in a precise manner, but only in an averaged way. The medium is then entirely characterized by a unique control parameter E/Γ, and an
J. Phys. Chem., Vol. 100, No. 30, 1996 12319 analytical expression of the superexchange coupling VS has been given. When the intervening medium is under a “wide-band” regime, its microscopic parameters may greatly influence the exact superexchange coupling, but this latter may still be relatively well understood on the basis of the VMA. However, for such ET systems, any calculation will be limited by the very large sensitivity of VS to the chosen parameters. Before our conclusions could reasonably be extended to real protein systems, it is necessary to take into account the distinction between covalent and noncovalent intersite couplings. Only the latter have been included in the present study, thus accounting for the three-dimensional topology of the intervening medium. Inclusion of covalent interactions introduces the onedimensional topology of the protein, together with an enhanced sensitivity of the electronic coupling to the details of the protein’s structure. Although the simultaneous inclusion of both “through-space” and “through-bond” interactions is the object of a forthcoming paper,23 it is worth mentioning here a major result of that work which is relevant to the application of our present study to proteins. In fact, we have shown that the contributions to ET of both types of couplings are mixed in an intricate way, so that only under exceptional energetic conditions may one dominate over the other. In such ET systems, neither a uniform barrier tunneling model2 nor a dominant pathway model5 can account for the full behavior of the effective electronic coupling VS. Finally, we should also note that through-space interactions cannot be accounted for in a perturbative way,23 so that the concept of an electronic band generated by the noncovalent couplings remains particularly useful even in the presence of covalent bonds. Acknowledgment. The authors thank Dr. Atsuo Kuki for stimulating discussions. They are also grateful to the Medical Research Council of Canada for support of this research. References and Notes (1) See, for example: (a) Electron Transfer in Inorganic, Organic, and Biological Systems; Bolton, J. R., Mataga, N., McLendon, G., Eds.; Advances in Chemistry Series 228; American Chemical Society: Washington, DC, 1991. (b) Long-Range Electron Transfer in Biology; Palmer, G. A., Ed.; Structure and Bonding, Vol. 75; Springer-Verlag: Berlin, 1991. (c) Electron Transfer Reactions in Metalloproteins; Sigel, H., Sigel, A., Eds.; Metal Ions in Biological Systems, Vol. 27; Marcel Dekker: New York, 1991. (d) Electron Transfer Reactions. Chem. ReV. 1992, 92, 365-490. (e) Electron Transfer. Chem. Phys. (Special Issue) 1993, 176, 289-629. (f) Lippard, S. J.; Berg, J. M. Principles of Bioinorganic Chemistry; University Science Books: Mill Valley, CA, 1994. (g) Gray, H. B.; Ellis, W. R., Jr. Electron Transfer. In Bioinorganic Chemistry; Bertini, I., Gray, H. B., Lippard, S. J., Valentine, J. S., Eds.; University Science Books: Mill Valley, CA, 1994; p 315. (2) Moser, C. C.; Keske, J. M.; Warncke, K.; Farid, R. S.; Dutton, P. L. Nature 1992, 355, 796. Moser, C. C.; Dutton, P. L. Biochim. Biophys. Acta 1992, 1101, 171. Farid, R. S.; Moser, C. C.; Dutton, P. L. Curr. Opin. Struct. Biol. 1993, 3, 225. Dutton, P. L.; Moser, C. C. Proc. Natl. Acad. Sci. U.S.A. 1994, 91, 10247. (3) (a) Onuchic, J. N.; Beratan, D. N.; Winkler, J. R.; Gray, H. B. Annu. ReV. Biophys. Biomol. Struct. 1992, 21, 349. (b) Wuttke, D. S.; Bjerrum, M. J.; Winkler, J. R.; Gray, H. B. Science 1992, 256, 1007. (c) Casimiro, D. R.; Wong, L.-L.; Colo´n, J. L.; Zewert, T. E.; Richards, J. H.; Chang, I-J.; Winkler, J. R.; Gray, H. B. J. Am. Chem. Soc. 1993, 115, 1485. (d) Casimiro, D. R.; Richards, J. H.; Winkler, J. R.; Gray, H. B. J. Phys. Chem. 1993, 97, 13073. (e) Karpishin, T. B.; Grinstaff, M. W.; KomarPanicucci, S.; McLendon, G.; Gray, H. B. Structure 1994, 2, 415. (4) (a) Farver, O.; Skov, L. K.; Pascher, T.; Karlsson, B. G.; Nordling, M.; Lundberg, L. G.; Va¨nnga˚rd, T.; Pecht, I. Biochemistry 1993, 32, 7317. (b) Langen, R.; Chang, I-J.; Germanas, J. P.; Richards, J. H.; Winkler, J. R.; Gray, H. B. Science 1995, 268, 1733. (5) (a) Beratan, D. N.; Betts, J. N.; Onuchic, J. N. Science 1991, 252, 1285. (b) Beratan, D. N.; Onuchic, J. N.; Winkler, J. R.; Gray, H. B. Science 1992, 258, 1740. (c) Regan, J. J.; Risser, S. M.; Beratan, D. N.; Onuchic, J. N. J. Phys. Chem. 1993, 97, 13083. (d) Skourtis, S. S.; Regan, J. J.; Onuchic, J. N. J. Phys. Chem. 1994, 98, 3379.
12320 J. Phys. Chem., Vol. 100, No. 30, 1996 (6) (a) Gamow, G. Z. Phys. 1928, 51, 204. (b) DeVault, D. QuantumMechanical Tunnelling in Biological Systems, 2nd ed.; Cambridge University Press: Cambridge, 1984. (7) Ulstrup, J. Charge Transfer Processes in Condensed Media; Lecture Notes in Chemistry, Vol. 10; Springer-Verlag: Berlin, 1979. (8) Economou, E. N. Green’s Functions in Quantum Physics, 2nd ed.; Springer-Verlag: Berlin, 1990. (9) Lopez-Castillo, J.-M.; Filali-Mouhim, A.; Plante, I. L.; Jay-Gerin, J.-P. J. Phys. Chem. 1995, 99, 6864. (10) Evenson, J. W.; Karplus, M. Science 1993, 262, 1247. (11) See, for example: (a) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (b) Simulation of Liquids and Solids. Molecular Dynamics and Monte Carlo Methods in Statistical Mechanics; Ciccotti, G., Frenkel, D., McDonald, I. R., Eds.; North-Holland: Amsterdam, 1987. (12) Beratan, D. N.; Onuchic, J. N.; Gray, H. B. Electron Transfer Reactions in Metalloproteins; Sigel, H., Sigel, A., Eds.; Metal Ions in Biological Systems, Vol. 27; Marcel Dekker: New York, 1991; p 97. (13) Gruschus, J. M.; Kuki, A. J. Phys. Chem. 1993, 97, 5581. (14) Siddarth, P.; Marcus, R. A. J. Phys. Chem. 1990, 94, 8430; 1993, 97, 2400, 6111, 13078. Siddarth, P. J. Photochem. Photobiol. A: Chem. 1994, 82, 117. (15) Stuchebrukhov, A. A.; Marcus, R. A. J. Phys. Chem. 1995, 99, 7581.
Filali-Mouhim et al. (16) Elliott, R. J.; Krumhansl, J. A.; Leath, P. L. ReV. Mod. Phys. 1974, 46, 465. (17) Newton, M. D. Chem. ReV. 1991, 91, 767. (18) Evenson, J. W.; Karplus, M. J. Chem. Phys. 1992, 96, 5272. See also: McConnell, H. M. J. Chem. Phys. 1961, 35, 508. Beratan, D. N.; Hopfield, J. J. J. Am. Chem. Soc. 1984, 106, 1584. (19) See, for example: Reichl, L. E. A Modern Course in Statistical Physics; University of Texas Press: Austin, 1980. (20) Wolfsberg, M.; Helmholz, L. J. Chem. Phys. 1952, 20, 837. (21) Lopez-Castillo, J.-M.; Tremblay, A.-M. S. Phys. ReV. B 1986, 34, 8482. (22) We would like to correct here some numerical errors that we have found in I. In fact, the seventh and eighth sentences of the next to last paragraph of section VI (“Discussion and Conclusions”) should be modified to read: “Taking the Evenson and Karplus23 numerical data (E ) 4 eV, V0 ) -6 eV, β ) 1.7 Å-1, using our notations) with Fβ-3 ) 0.01, we find E/Γ ) -2.6 yielding a value βCMA ) 1.05 from eq 37 (which corresponds to the electron-transfer case). The exact numerical calculation on the structure of tuna cytochrome c gives a slope of 1.11, which is remarkably close to the CMA prediction”. (23) Lopez-Castillo, J.-M.; Jay-Gerin, J.-P. J. Phys. Chem., in press.
JP9535147
