Matrices quantum exchange coupling

A quantum mechanical treatment combined with the density matrix formalism extends the description to include the dynamic spectra of spin-coupled systems. (34-38) Further developments in the theory and presentation thereof, in a form suitable for computer calculations, are due to Binsch et al., and to Kaplan et al. (14, 15, 39) However, even the recent theories are not rigorous in certain aspects and contain some errors. This is particularly true in the case of the intermolecular exchange of spins. 

Classical, semi-classical, and quantum mechanical procedures have been developed to rationalize and predict the rates of electron transfer. In summary, the observed rate of a self-exchange reaction can be calculated as a function of interatomic distances, force constants, electronic coupling matrix element, and solvent parameters. These model parameters are either calculated, estimated, or determined by experiment, in each case with a corresponding standard deviation. Error propagation immediately demonstrates that calculated rates have error ranges of roughly two orders of magnitude, independent of the level of sophistication in the numerical procedures. 

The first quantum-mechanical consideration of ET is due to Levich and Dogonadze [7]. According to their theory, the ET system consists of two electronic states, that is, electron donor and acceptor, and the two states are coupled by the electron exchange matrix element, V, determined in the simplest case by the overlap between the electronic wave functions localized on different redox sites. Electron transfer occurs by quantum mechanical tunneling but this tunneling requires suitable bath fluctuations that bring reactant and product energy levels into resonance. In other words, ET has... 

The exchange interaction (the last term in the above equation), which couples states of different arrangements, is analogous to electron exchange interactions in quantum chemistry that arise from matrix elements in which the electron coordinates have been permuted (i.e. exchanged). The coupled... 

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