A non-adiabatic theory of Electron Transfer (ET), which improves the standard theory near the inversion point and becomes equivalent to it far from the inversion point, is presented. The complex amplitudes of the electronic wavefunctions at different sites are used as Kramers variables for describing the quantum tunneling of the electron in the deformable potential generated by its environment (nonadiabaticity) which is modeled as a harmonic classical thermal bath. After exact elimination of the bath, the effective electron dynamics is described by a discrete nonlinear Schrödinger equation with norm preserving dissipative terms and a Langevin random force, with a frequency cut-off, due to the thermalized phonons.This theory reveals the existence of a specially interesting marginal case when the linear and nonlinear coefficients of a two electronic states system are appropriately tuned for forming a Coherent Electron-Phonon Oscillator (CEPO). An electron injected on one of the electronic states of a CEPO generates large amplitude charge oscillations (even at zero temperature) associated with coherent phonon oscillations and electronic level oscillations. This fluctuating electronic level may resonate with a third site which captures the electron so that Ultrafast Electron Transfer (UFET) becomes possible. Numerical results are shown where two weakly interacting sites, a donor and a catalyst, form a CEPO that triggers an UFET to an acceptor. Without a catalytic site, a very large energy barrier prevents any direct ET. This UFET is shown to have many qualitative features similar to those observed in the primary charge separation in photosynthetic reaction centers. We suggest that more generally, CEPO could be a paradigm for understanding many selective chemical reactions involving electron transfer in biosystems.