9266
J. Phys. Chem. 1993,97, 9266-9269
ARTICLES Superexchange Coupling and Non-Nearest-Neighbor Interactions in Electron Transfers J.-M. Lopez-Castillo, A. Filali-Mouhim, and J.-P. Jay-Gerin’ Groupe du Conseil de Recherches Midicales du Canada en Sciences des Radiations et Dipartement de Midecine Nucliaire et de Radiobiologie. Faculti de Midecine, Universiti de Sherbrooke. Sherbrooke, QuCbec Jl H 5N4,Canada Received: March 19, 1993; In Final Form: June 25, 1993’
Non-nearest-neighbor interactions in the problem of electron transfer via superexchange are considered. In particular, second-neighbor coupling in a finite disordered chain and exponential coupling in an infinite periodic chain are investigated in detail. Inclusion of these interactions is formulated through a renormalization of the energies as well as of the coupling matrix elements which are constructed by a simple recursive scheme. The form of the extended McConnell formula is then applicable. It is shown that, under certain conditions, the effective coupling between the donor and acceptor sites may suffer large changes.
Extensive theoretical work has recently been devoted to the understanding of the role of the intermediate medium in the determination of the electronic coupling TADbetween distant redox centers.l This coupling is a major factor involved in the rate of electron-transfer reactions2 occurring in a wide variety of systems,ranging from small chemical to huge biological molecules. For nonadiabatic reactions, TADis related to the electron-transfer rate constant k via the generalized version of Fermi’s Golden Rule
k = 3TAD12(FCWD) where h is Planck’s constant divided by 27 and (FCWD) is the so-called Franck-Condon-weighted density of states which accounts for the contribution of nuclear coordinates to the kinetic process. The total electronic coupling TADbetween the acceptor (A) and donor (D) sites is the sum of the direct coupling VADand the elecyonic superexchange coupling VSwhich is the quantity of interest in this work. The simple one-electron,tight-binding Hamiltonian is used as the starting point of most theoretical calculations. So far, a few analytical results have been obtained for the case of nearestneighbor interactions, starting with the work of McConnell,’ who gave the expression
for a one-dimensionalbridge consisting of N identical units. In eq 2, Tis the coupling of A and D to the bridge, t is the coupling between two adjacent units in the bridge, and E is the energy gap between the redox sites and the bridge. McConnell’s formula has then been generalized to the case of aperiodic bridging chains presenting both diagonal and off-diagonal disorder so that1
sites, and VDand VAdenote the couplingsof D and A to the chain ends, respectively. Equations 2 and 3 give the contribution of one single electron-transfer pathway to VS. Within the nearestneighbor approximation, it has been shown that, for situations in which multiple bridge pathways can occur between donor and acceptor, the exact expression of V. assumes the same form as in eq 3, where the single-pathway energy parameters Ei are now replaced by a renormalized expre~sion.~ From recent work it is becoming clear that inclusion of nonnearest-neighbor interactions, and consequently of many interfering pathways, is necessary to describe the effective electronic coupling in many molecular systems.5 It is the purpose of the present paper to show how this inclusion may be carried out so that a modified form of McConnell’s formula can be obtained. The present tight-binding model Hamiltonian for a disordered one-dimensional chain containing N sites is (4) where Ei is the energy of the ith site, and the coupling matrix elements tij between sites i and j are
indicating that only fmt- and second-nearest-neighborinteractions are retained. This Hamiltonianappliesto a single one-dimensional chain, such as that depicted in Figure la, as well as to any chain inwhich thenumberingofatomsintheorder 1’2, ...,Nisconsistent with the coupling of eq 5 . A double chain, such as that depicted in Figure 1b, which may be viewed as corresponding to a single one-dimensional chain with two orbitals per site, may also be described in this way. Vs takes the form4 V, = V~G”V,
(3) where where tit+l is a nearest-neighbor coupling matrix element of the chain, Ei is the energy gap between the bridge site i and the redox *Abstract published in Aduance ACS Absrracrs, August 15, 1993.
0022-3654/93/2097-9266$04.00/0
GN = (Eo- HN)-’ (7) is the Green’s function6for the medium, calculated at the energy EOof the donor and acceptor (EO= 0 is taken as the reference), 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 37, 1993 9267
Superexchange and Non-Nearest-Neighbor Interactions
10‘~
lo4 146
I24
b)
1o-6
io-’
Figure 1. Chain models for the Hamiltonian of e q s 1 and 2: (a) single
chain with one orbital per site and (b) double chain or single chain with two orbitals per site. To see how the latter chain can be viewed in this way, consider labels 1 and 2 as representing the two orbitals of a single site and similarly for the following pairs of labels. Then, the first- and second-neighbor couplings represent intrasite and intersite orbital couplings, respectively.
>*
‘’
I,
I
I
I
0.0
0.1
0.2
0.3
0.4
0.0
0.1
0.2
0.3
0.4
lo-*
and VD and VAdenote the couplings of D and A to the chain ends, respectively. Because of the additive form of eq 6, it is sufficient to calculate Vs for the case of a coupling of the donor and acceptor to the “terminal” sites 1 and N of the medium, respectively. Equation 6 is then written as 1U
V, = crN (8) where we have taken VD = VA = 1 without loss of generality. Note that VSis just the matrix element l , N of the inverse matrix of (-If?. Formally, eq 8 may be rewritten as7 ‘:3 V, = Gf2-3‘12
GN-1 1N-1 ‘?N GN-2 G N - ~
...-1N-2
(9)
IN-1
with the matrix elements of the inverse matrix given by
G:i
(-l)i- 4
Second-neighbor coupling t’ Figure 2. Effective superexchange coupling lVsl as a function of the second-nearest-neighborcoupling strength t’for a chain of 10sites. Each
curve uses a singleset of randomly generated energies and couplings. The energies areuniformlydistributedin theinterval [1,1.5], thefirst-nearestneighbor couplings are uniformly distributed (a) in the interval [0.2,0.5] and(b) in the interval [0.4,0.7], and thesecond-nearest-neighbor couplings are uniformly distributed in the interval [t’,t’+ 0.051. - - -,Exact results from the numerical inversion of HN;-, approximate results from eqs 13-15. Energies and couplings are in arbitrary units.
i
t12
v,= ( - 1 ) N
IW
where the symbols A and I I are used to denote the first minor and the determinant, respectively. In the special case of the Hamiltonian of eq 4, we find that Afi and IHl obey the following recurrence relations
+ ...
~ - 1
E1E2- ti2 i=2
(14)
ci
where the factor [tt$/(EIE2- ti2)] accounts for the initial (or first) link of the chain, and Ci
AKll = tii+lAfi- ri+li-lEiAi;!l
n
ti,+, =-
E,+,
(1 l a )
For an application of this relation, the renormalized coefficients must be computed iteratively using the recurrence relations (13) with the starting values .2
where the nearest-neighbor interaction is taken exactly into account and where we have neglected terms involving products of second-neighbor interactions. From eqs 10 and 11, we find
Now, defining the renormalized parameters recursively
-tii+l = tii+l - Ei 7 ti-, i+ 1
(13a)
ti-li
and
Vs takes the familiar form of an extended McConnell formula:
8, = E l ,
t12
=4
2
The importance of eqs 13;15 becomes clear when one realizes that the effective coupling rWl may strongly be affected by small values of the second-neigh_borcoupling tclf+l. This is because of the weighting factor (Ei/ri-li)in eq 13a, which is expected to be important for nonresonant electron transfer. This effect is illustrated in Figure 2 where we compute VS for a disordered chain consisting of 10 sites as a function of the second-neighborcoupling strength t ’. Parts a and b of Figure 2 have been obtained using first-neighbor couplings uniformly distributed in the intervals [0.2,0.51 and [0.4, 0.71,respectively. As we can see from these figures, the exact result (obtained from the direct numerical inversion of the matrix HN)is recovered a t vanishing t’. Figure 2a shows that the prediction of eqs 13-15 agrees quantitatively with the exact result up to t’ = 0.1 and only approximately a t higher coupling values. A similar qualitative agreement is observed in Figure 2b though in that case quantitative agreement no longer exists. In both cases, and in all others studied but not shown here, the discrepancy between the exact and
9268 The Journal of Physical Chemistry, Vol. 97, No. 37, 1993
Lopez-Castillo et al. 5
0.0
0.2
0.6
0.4
0.8
1.0
:
’
’
”
O l ’ ”
I
0.2
0.0
approximate results is observed for coupling values which are no longer negligible as compared to the chosen energy gap (see the legend of Figure 2). In such a situation, the system is brought close to the resonance condition and hence to the regime of adiabatic electron transfer. This discrepancy is thus not a limitation of the renormalized-couplingconcept introduced here since the overall formalism used in nonadiabatic electron-transfer theory already assumes that we are far from the resonance condition. The important result illustrated in Figure 2 is the drastic effect of this second-neighborcouplingwith the vanishing of the total superexchangecouplingdue to interferences between pathways of equal importance involving both first- and secondnearest neighbors. Equations 13-1 5 obviously give the known exact result4 in the limit of a simple nearest-neighbor interaction. In particular, for the infinite periodic chain (El = E, tn+l = t), we find
As is well-known, eq 17 corresponds to an exponential decay as a function of the chain length characterized by the effectiveinverse decay length
Bcr is represented in Figure 3, along with the McConnell and WKB approximations, as a function of x. Although these approximations work in the limits x = 0 and x = 1, respectively! it is surprising (see Figure 3) that McConnell’s approximation is good up to x = I/* while the WKB approximationworks beyond this value. Inclusion of non-nearest-neighborinteractions beyond the first order in the perturbative approach presented above is not difficult but tedious. It is interesting to note, however, that the infinite periodic chain allows for an exact calculation of Vs in the more complex case of an exponential coupling between the medium sites, as in the extended Hiickel Hamiltonian. We write
with
where we have taken a spacing between sites equal to unity so that 9 , is the exponential decay coefficient per site and t is the coupling between nearest neighbors. Diagonalization of H is
“
’
.. -
“ ! I
“
.
0.4
0.6
0.8
1.0
12vEI
2m
Figure 3, Effective inverse decay length Bfi as a function of Z f / E .The exact results of eqs 17 and 18 are compared to the McConnell [BMC= -In (lf/4)] and WKB [@- = (2mEg/h2)1/2] approximations. Here, m is the effective mass and Eg is the energy gap. These results apply to infinite periodic chains with only nearest-neighbor interactions.
”
Figure 4. Effective inverse decay length Paas a function of 12t/hl. The exact results of eqs 24 and 25 are compared to the nearest-neighbor approximationfor the infiite periodicchain described by the Hamiltonian of eqs 19 and 20 with e+ = 0.1. Note that Ben is infinite at 2 f / E = 0.2. ---,Nearest-neighbor approximation;-, exact ( 2 t / E > 0);- - -,
..
exact ( 2 f / E < 0).
straightforward, and we get the eigenvalues ck
=E
+ 2f(
cos(k) -e+ 1 - 2e” cos(k) e-28
+
while the eigenstates are simply plane waves
+k(n) = elkn
- T I ~ < T (22)
The effective superexchange coupling VSis now given by
which may be calculated analytically. We find an exponential decay law as a function of N with given by
with the renormalized coefficients
B = ~ ( +1e-’fl)
- te-8
(29))
Equation 24 is interesting in that it takes the same form as eqs 17 and 18 corresponding to the nearest-neighbor model. In particular, the same discussion applies to the limiting cases R = 0 and R = 1, corresponding to the McConnell and WKB approximations,respectively. The effective inverse decay length BCrof eq 24 is plotted in Figure 4, along with the prediction of the nearest-neighbor approximation of eq 18, as a function of 12t/4 for e-8 = 0.1 which represents the ratio of the second- to the first-nearest-neighbor couplings. Although the above derivation applies to an infinite chain, we verified numerically that finite size effects are in fact negligible. An important feature of the present result is that /3dbecomes infinite when x = 2e.B. This result is confirmed numerically for finite chains of length N > 2 (see Figure la). In Figure 5 , it is seen that a sudden decrease of JVs(occurs on passing from 2 to 3 atoms in the chain when t = Ee-8. It is worth noting that the renormalized coefficients are not uniquely defined, depending on the order by which decimation is performed. For example, starting with the chain end instead of the beginning yields different renormalized energies as is readily seen with the N = 2 case. The reason for this nonuniqueness is that the renormalized coefficients are nonlocal properties of the
Superexchange and Non-Nearest-Neighbor Interactions
1
N
Figure5. Effective superexchangecoupling for a periodic chain consisting of N sites described by the Hamiltonian of eqs 19 and 20. Parameters correspond to the singularity of Figure 4. The sudden jump on passing from N = 2 to N = 3 agrecs with the prediction made for the infinite chain. Note that the residual coupling after N = 3 only corresponds to numerical noise since the coupling is infinitely small. Vs is in arbitrary units.
chain. Moreover, this leads to theconcept of an effectivepathway that depends on the numbering order of the sites in the chain. For the case of the chain depicted in Figure lb, the chosen labeling defines a pathway whereno previous interactions existed between consecutive sites (such as 2-3, 4-5, etc.) so that it must be considered as unphysical. Nevertheless, the renormalization concept presents the advantage of recovering the familiar form of the extended McConnell formula. The present generalization, along with the known extension to simple side groups whose effect reduces to a further renormalization of the energies? gives a wide range of applicability to it. In conclusion, it has been shown that inclusion of non-nearestneighbor interactions in the one-dimensional chain model can be made by introducing renormalized energy and nearest-neighbor coupling coefficients. An explicit method consisting of a simple recursive scheme has been presented. The concept of a renormalized coupling is new and should be used instead of the bare coupling in the study of actual one-dimensional electron-transfer systems. We find that, under quite ordinary conditions, electron
The Journal of Physical Chemistry, Vol. 97, No. 37, 1993 9269 transfer may be drastically reduced by second-neighbor coupling so that two consequences can be deduced. These are (i) by controlling the energy of a single site, electron transfer may be changed from easy to impossible,I0which is a condition for an efficient electronic switch at the molecular level, and (ii) when discrimination between many pathways is considered in threedimensional systems such as proteins, non-nearest-neighbor interactions should be taken into account. Investigation of nonnearest-neighbor interactionsin more complex, three-dimensional model systems is presently under way. Acknowledgment. The authors wish to thank Dr. Atsuo Kuki for introducing them to this subject and Professor Hans Bichsel for a critical reading of the manuscript. They are also grateful to the Medical Research Council of Canada for support of this research. References and Notes (1) A number of articles and reviews have recently appeared. See,for example: Newton, M. D. Chem. Rev. 1991,91,767. Ratner, M. A. J. Phys. Chem. 1990, 94,4877. (2) The following books give an overview of the field: (a) Long-Range Electron Transfer in Biology; Structure and Bonding 75; Springer-Verlag: Berlin, 1991. (b) Electron Transfer Reactions in Metalloproteins; Sigel, H., Sigel, A., Eds.; Metal Ions in Biological Systems 27; Marcel Dekker: New York, 1991. (c) Electron Transfer in Inorganic, Organic, and Biological Sysrems; Bolton, J. R., Mataga, N., McLendon, G., Eds.; Advanm in Chemistry Series 228; American Chemical Society: Washington, DC, 1991. (d) Electron Transjer in Biology and the Solid State; Johnson, M. K., King, R. B., Kurtz, D. M., Jr., Kutal, C.,Norton, M. L.,Scott, R. A., Eds.;Advances in Chemistry Series 226; American ChemicalSociety: Washington, DC, 1990. (3) McConnell, H. M. J. Chem. Phys. 1961, 35, 508. (4) Da Gama, A. A. S . J. Theor. Biol. 1990, 142, 251. (5) See,for example: (a) Naleway, C. A.; Curtiss, L. A,; Miller, J. R. J . Phys. Chem. 1991,95,8434. (b) Liang, C.; Newton, M. D. J . Phys. Chem. 1992, 96, 2855. (c) Jordan, K. D.; Paddon-Row, M. N. Chem. Reu. 1992, 92, 395. (6) Economou, E. N. Green’s Functions in Quantum Physics, 2nd ed.; Sarineer: New York. 1983. (7) Onuchic, J. N.;de Andrade, P. C. P.; Beratan, D. N. J . Chem. Phys. 1991, 95, 1131. (8) Beratan, D. N.; Onuchic, J. N.; Hopfield, J. J. J. Chem. Phys. 1985, 83, 5325. (9) Onuchic, J. N.; Beratan, D. N.; Winkler, J. R.; Gray, H. B. Annu. Rev. Biophys. Biomol. Struct. 1992, 21, 349. (10) Goldman, C. Phys. Rev. A 1991, 43, 4500.
