  The complex amplitudes of the electronic wavefunctions on different sites are
used as Kramers variables for describing Electron Transfer. The strong coupling
of the electronic charge to the many nuclei, ions, dipoles, etc, of the
environment, is modeled as a thermal bath better considered classically. After
elimination of the bath variables, the electron dynamics is described by a
discrete nonlinear Schrodinger equation with norm preserving dissipative terms
and Langevin random noises (at finite temperature). The standard Marcus results
are recovered far from the inversion point, where atomic thermal fluctuations
adiabatically induce the electron transfer. Close to the inversion point, in
the non-adiabatic regime, electron transfer may become ultrafast (and
selective) at low temperature essentially because of the nonlinearities, when
these are appropriately tuned. We demonstrate and illustrate numerically that a
weak coupling of the donor site with an extra appropriately tuned (catalytic)
site, can trigger an ultrafast electron transfer to the acceptor site at zero
degree Kelvin, while in the absence of this catalytic site no transfer would
occur at all (the new concept of Targeted Transfer initially developed for
discrete breathers is applied to polarons in our theory). Among other
applications, this theory should be relevant for describing the ultrafast
electron transfer observed in the photosynthetic reaction centers of living
cells.
