A recursion relation is formulated for the Green’s function for calculating the effective electron coupling in bridge-assisted electronic transfer systems, within the framework of the tight-binding Hamiltonian. The recursion expression relates the Green’s function of a chain bridge to that of the bridge that is one unit less. It is applicable regardless of the number of orbitals per unit. This method is applied to the system of a ferrocenylcarboxy-terminated alkanethiol on the Au(111) surface. At larger numbers of bridge units, the effective coupling strength shows an exponential decay as the number of methylene(–CH2–) units increases. This sequential formalism shows numerical stability even for a very long chain bridge and, since it uses only small matrices, requires much less computer time for the calculation. Identical bridge units are not a requirement, and so the method can be applied to more complicated systems.

1.
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J. R.
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Science
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,
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(
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Experimental investigations of electron transfer across self-assembled monolayers on electrode surfaces include those of
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11.
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236
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L. S.
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the single band problem is also solved and discussed in detail by
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15.
S. G. Davison and M. Stȩślicka, Basic Theory of Surface States (Clarendon, Oxford, 1992).
16.
The overlap integrals Sij of the atomic orbitals in tight-binding model were treated in the following way: (1) The overlap integrals were only nonzero within each CH2 bridge unit. (2) Within each CH2, the molecular orbitals are obtained by solving the linear equation hp = eSp for e, and p, where h is the Hamiltonian, S is the overlap integral, e, is a diagonal matrix with the eigenvalues as its diagonal matrix elements, and p is composed of column eigenvectors (p = (v⃗1,…,v⃗m)). Solving the above equation is equivalent to solving the usual eigenvalue and eigenvector problem hp = ep, where h = S−1/2hS−1/2 and p = S1/2pS1/2 = S1/2(v⃗1,…,v⃗m)S1/2, and p forms an orthonormal basis set with which Eq. (8) is written.
17.
The first non-zero contribution of G(1,n)(n) in the perturbative expansion of Eq. (12) is the first-order term G0(n)H1(n)G0(n). With Eqs. (13) and (18) the first-order contribution to G(1,n)(n) can be obtained as G(1,n−1)(n−1)vΔ−1. So G(1,n)(n)≈G(1,2)(2)(vΔ−1)n−2. This expression is similar to Eq. (3), which is McConnell’s estimation.
18.
If we denote G(1,n)(n) = G(1,2)(2)xn with xn≡∏i = 3nvΔ−1Ni, then xn has solution xn = [α1n−3m1−α2n−3m2]−1, for n⩾3 with α1 = Δv−1/2 + δ, α2 = Δv−1/2−δ, m1 = α1δ−1α1/−δ−1α1vTΔ−1N2/2, m2 = α2δ−1α2/2−δ−1α2vTΔ−1N2/2, and δ is a matrix that satisfies δ2 = −vTv−1+(Δv−1)2/4. It is seen that upon matrix inversion to obtain xn, the eigenvalue that is the smallest in modulus of 1n−3m1−α2n−3m2] is the most important, while in numerical calculation of α1,2n−3 for large n, the smaller eigenvalues of α1, are usually not preserved. In the present form, a specially designed algorithm would be needed in any numerical calculation using above expressions.
19.
The factor β is conventionally defined as either of the following: kET∝e−βd, or |V¯|2∝e−βd (Eq. (30)). They are slightly different because the reorganizational energy λ is dependent of distance. Here, the latter definition was used, with d being taken as the number of methylene units.
20.
For integrating over the electronic energy levels in the metal the weighting function in Eq. (6) of
R. A.
Marcus
,
J. Chem. Phys.
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See AIP Document No. PAPS JCPSA-106-584-19 for 19 pages of a list of the coordinate used for the calculation and a figure showing the geometry of the principal chain and a neighboring chain.
Order by PAPS number and journal reference from American Institute of Physics, Physics Auxiliary Publication Service, Carolyn Gehlbach, 500 Sunnyside Boulevard, Woodbury, NY 11797-2999. Fax: 516-576-2223, e-mail paps@aip.org. The price is $1.50 for each microfiche (98 pages) or $5.00 for photocopies of up to 30 pages, and $0.15 for each additional page over 30 pages. Airmail additional. Make checks payable to the American Institute of Physics.
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The calculation of effective coupling |HDk|2¯ does not depend on λ (Eqs. (43) and (44)), so the calculation for various of chain lengths is a straight-forward generalization without considering the dependence of λ on the chain length n. For calculating the reaction rate with Eq. (30), proper values of λ are needed for different chain lengths, which has been discussed in the following reference:
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