Extended weak-coupling limit

The material in this chapter is largely organized around the molecular properties that contribute to electron transfer processes in simple transition metal complexes. To some degree these molecular properties can be classified as functions of either (i) the nuclear coordinates (i.e., properties that depend on the spatial orientation and separation, and the vibrational characteristics) of the electron transfer system or (ii) the electronic coordinates of the system (orbital and spin properties). This partitioning of the physical parameters of the system into nuclear and electronic contributions, based on the Born-Oppenheimer approximation, is not rigorous and even in this approximation the electronic coordinates are a function of the nuclear coordinates. The types of systems that conform to expectation at the weak coupling limit will be discussed after some necessary preliminaries and discussion of formalisms. Applications to more complex, extended systems are mentioned at the end of the chapter. 

The principal role of diffusion in these processes could be established considering rather simple examples [2]. If the kinetic equations for a well-stirred system are able to reproduce self-oscillations (the limit cycle), the extended system could be presented as a set of non-linear oscillators continuously distributed in space. Diffusion acts to conjunct these local oscillations and under certain conditions it can result in the synchronisation of oscillations. Thus, autowave solutions could be interpreted as a result of a weak coupling (conjunction) of local oscillators when they are not synchronised completely. The stationary spatial distributions in an initially homogeneous systems can also arise due to diffusion, which makes homogeneous solutions unstable. 

SXAPS and AEAPS spectra may differ largely because of the core hole decay mechanisms following the excitation of the core electrons in these spectroscopies. In SXAPS X-ray emission is slow and core hole production and de-excitation are only weakly coupled. On the other hand, in AEAPS Auger decay is fast and the excitation and decay are strongjy coupled. This may lead to some broadening of structure in AEAPS. Dose et al. [94] have observed in solid Ni the smearing of the threshold slope and structure in the AEAPS spectrum as compared to the SXAPS spectrum. APS is, however, not limited to solid metals only. With proper experimental arrangement it could be extended to the study of liquid metals, as has been demonstrated by Dose et al. [94]. 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

We have presented several approaches to calculate the rate constants of electron transfer occurring in solvent from the weak to strong electronic couplings. In the fast solvent relaxation limit, the approach based on the nonadiabatic transition state theory can be adopted. It is related to the Marcus formula by a prefactor and referred as a modified Marcus formula. When the solvent dynamics begin to play a role, the quantum Kramers-like theory is applicable. For the case where the intramoleeular vibrational motions are much faster than the solvent motion, the extended Sumi-Mareus theory is a better ehoice. As the coherent motion of eleetron is ineorporated, such as in the organic semiconductors, the time-dependent wavepaeket diffusion approach is proposed. Several applications show that the proposed approaches, together with electronic structure calculations for the faetors eontrolling eleetron transfer, can be used to theoretically predict electron transfer rates correctly. 

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