Vibrational Model for ET in the Limit of Low Barrier

Completely delocalized ionization is obtained in the coqjngated case when only one single bond separates the donble bonds. More than one separating bond leads to mode softening and partial localization, whereas a completely localized, ionized double bond is obtained if many single bonds separate the donble bonds. 

0 is a function of Q and both eigenfunctions are normalized, assuming )a and 1) are normalized. 

These are the ordinary Marcus curves for a symmetric systan. The ground state forms a symmetric double-well potential. The BO approximation does not work here because if we are at one minimum, the electron is at the donor D and if we are at the other minimum, the electron is at the acceptor A. 

Previously, we treated the spectrum as being either localized or delocalized (single molecule). To solve the problem accurately, it is necessary to go beyond the BO approximation  

Xk is a set of harmonic functions centered at Q = -Qq, and Q = Qo (lower) and Q = 0 (upper). Equation 10.21 gives the BO energy surfaces. From the BO approximation, we also obtain equations for the vibrations. We insert T from Equation 10.99 into these equations and obtain 

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