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A relationship between electron transfer rates and molecular conduction

This note discusses the relationship between a given intramolecular bridge assisted electron transfer rate and the corresponding zero bias molecular conduction of the same molecular species.


Introduction
Molecular electron transfer, one of the most basic chemical processes, has been an active field of research for over half a century. [1][2][3][4][5][6] Investigations of this reaction on the fundamental level focus on the rate of the transfer process between donor and acceptor species that exist in solution either as free solutes or as separate sites of a bigger molecule. In addition to rates, the yield of an electron transfer reaction is a meaningful observable when competing processes exist. Theoretical studies of these reactions aim to understand the interplay between solvent dynamical properties and between molecular structure and dynamics in determining these rates and yields.
Another manifestation of molecular electron transfer is gaining increasing attention in recent years: the passage of electronic current in a nano-junction comprised of two metal leads and a connecting molecule or a molecular layer, where the molecule has the role of a conducting component (See,. This leads to a new type of measurement where the observable is the molecular conduction (ideally of a single molecule), or more generally, the current-voltage characteristic of the molecular junction. On the theoretical side, these observables (and their dependence on the molecular structure and the molecule-metal bonding) are usually described using variants of the Landauer formula. [10][11][12] The latter is a formal relationship between the electron transmission coefficient associated with a given nano-structure (a property derived from scattering theory) and its conduction. The needed electron transmission properties of the molecular junction are computed following methodologies applied earlier in studies of 'conventional' molecular electron transfer, 13,14 using quantum chemical methods at different levels of approximation, or (for discussions of generic phenomenology) simples models, e.g., the super-exchange model 15 of the donorbridge-acceptor (DBA) system.
Clearly, the conduction property of a given molecular system and the electron transfer properties of the same system should be closely related. One should keep in mind that because of tunneling there is always an Ohmic regime near zero bias. Obviously this conduction may be extremely low, indicating in practice an insulating behavior. 16 Of particular interest is to estimate the electron transfer rate in a given donor-bridge-acceptor (DBA) system that will translate into a measurable conduction of the same system when used as a molecular conductor between two metal leads. An earlier attempt in this direction 17 was limited to a 1-dimensional model and has disregarded the Franck-Condon factor in the electron transfer rate. In this paper I study this issue and derive an approximate practical relationship between these observables.

Theoretical considerations
Consider a DBA system, with a bridge that consists of N identical segments (denoted 1,2,...N) with nearest neighbor coupling V B . For this case the non-adiabatic limit of electron transfer theory is usually valid, and the corresponding rate is given by 2 where V DA is the coupling between the donor (D) and acceptor (A) electronic states and is the thermally averaged, Franck Condon (FC) weighted density of nuclear states. In Eq.
where V D1 and V NA are the corresponding coupling matrix elements, and where 1 ( ) N G E is a matrix element of the bridge Green's function. In the tight binding approximation (nearest neighbor coupling) and in the weak coupling limit, where E n and V n,n+1 bridge energies and coupling elements, and where the second equality in (6)  Suppose now that the same DBA complex is used to connect between two metal contacts such that the 'donor' and 'acceptor' species are chemisorbed on the two metals, denoted 'left' (L) and 'right' (R) respectively. (In real molecular junctions these species correspond to the end groups of the molecular chain, but we continue to use the terms donor and acceptor in order to maintain the analogy to the corresponding electron transfer process).
We wish to calculate the zero bias conduction g of this junction and its relation to .
First note that the conduction process does not involve localized states of the electron on the donor or the acceptor, so the factor ¦ will not appear in g. We assume that at zero bias the metal Fermi energy E F lies in the gap between the HOMO and LUMO of the molecular bridge. In this case dephasing and energy losses arising from transient distortions of the bridge nuclear configuration are relatively small and will be disregarded. Assuming as before that states of the molecular complex are coupled to the metal only via the D and A orbitals, and that the latter are coupled only to their adjacent metal contacts, the conduction (at zero bias) can be derived from the weak coupling limit of the Landauer formula, or directly from the golden rule. It is given by 18 Here e is the electron charge and ( ) The bars above G 1N and the coupling elements correspond to the fact that their values for the chemisorbed molecule may be different from the corresponding values for the chemisorbed molecule. Eqs. (5), (7) and (8) and Γ Γ are of order ~1eV the error in the estimate of X 3 will be of order 1 and since far from resonance (which was assumed above) G 1N (E) does not depend strongly on the energy the error in taking X 1 =1 is similarly small. Eq. (9) then lead to

Discussion and conclusion
The following observations can be made. First, the appearance of  Eq. (12) provides a practical expression for making such estimates, but its use should be exercised with caution. This expression is a relation between the conduction of a molecular structure made of a bridge connecting two special chemical groups chemisorbed on corresponding metal leads and the electron transfer rate of the same molecular structure in solution, where the same special groups play the roles of donor and acceptor. Its derivation involves a drastic assumption that, apart from a uniform energy shift, the electronic structure of this molecular structure when connecting between the metal leads is not considerably different from the electronic structure of the corresponding species in solution.
We note that most recent calculations of molecular conduction rely on similar assumptions, using as input electronic structural parameters of the free molecule (e.g., assuming that the molecular bridge is electrically neutral at zero bias). Another point to emphasize is that the result (12) corresponds only to conduction at zero bias. An important attribute of experimental setups involving molecular conductors is the possibility to employ their finite bias conduction, which is not directly related to the corresponding electron transfer property.
Keeping these limitations in mind, Eq. (12) provides a useful approximate relationship between two extremely important molecular transport properties.