Electron tranport by hopping

The bridge-assisted electron transfer discussed above is a coherent quantum-mechanical process. Its tunneling nature is manifested by the exponentially 

Problem 16.9. Show that the matrix K has one zero eigenvalue, and that the corresponding eigenvector is the equilibrium state that satisfies Pi/Pj = Qx llEj -EP/ksTY 

In general, the time evolution inferred from such a model is characterized by A -H 1 characteristic times, or rate coefficients, associated with the nonvanishing eigenvalues of the kinetic matrix K. In our application, however, the energies of the donor and acceptor states 7 = 0, A - -1 are considerably lower than those of the bridge states. Specifically, we assume that the inequalities 

Barring this possibility, we proceed to evaluate the effective transfer rate A a d-We will use the steady-state flux method, now a classical version of the scheme developed in Section 9.5. In this approach we consider the steady-state obtained at long time under the restriction that state 0 is a constant source (Po = constant) while state N + 1 is a drain (Pn+i = 0)- We also limit ourselves to the case where all rate coefficients that are not associated with species 0 and 7V+1 are the same, denoted k. Equations (16.118), without the first and last equations, then lead to the N steady-state equations 

Here J is the flux through the system that drains population out of state 7V-I-1. At steady state this flux also satisfies 

---

The electrochemical and spectroscopic data indicates that sites on these polymers can communicate with each other, in the electron transfer sense, on a relatively short time scale and without the formation of stable mixed valence clusters. Electronic tranport via hopping or tunnelling and modulated by means of neighboring molecular group collisions would be consistent with these requirements. The relative molecular nonspecificity of this mechanism suggests that other polymeric materials would show similar effects and this has been seen for thin films of poly — (vinylferrocene) and poly — (nitrostyrene). 

---