In a molecular electronic device, a key parameter of interest is the electronic coupling linking the various functional units (e.g. organic molecules/fragments, inorganic complexes, nano-electrodes). In practical applications, rate constants for specific electron-transfer processes can easily be evaluated considering all interactions within the system; often, exponential decreases in rate constants with increasing separation between functional units (bridge length) is found, and there is now considerable interest in finding systems with much slower, non-exponential decreases. The simple Hamiltonian model, first introduced in 1961 by McConnell, is qualitatively descriptive of most electron transfer processes and, with the aid of numerous analytical approximations, predicts an exponential bridge-length dependence. However, the range of validity of the approximations used is small and exponential falloff is known to be much more general and robust. We investigate the analytical solution recently obtained by Evenson and Karplus for various aspects of the problem and find that the most serious approximations used in analysing the McConnell Hamiltonian modify the value of the exponent rather than introduce non-exponentiality. Hence, we introduce some simple improved rate laws appropriate to both the exponential and non-exponential regimes. Also, the analysis is extended to consider important systems bridged byand/orbonds, in which the bridge band structure is more complex: similar rate laws are found to apply, and indeed all results obtained are expected to be generally descriptive.
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