Mixed quantum-classical dynamics on the exact time-dependent potential energy surface: A fresh look at non-adiabatic processes

The exact nuclear time-dependent potential energy surface arises from the exact decomposition of electronic and nuclear motion, recently presented in [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)]. Such time-dependent potential drives nuclear motion and fully accounts for the coupling to the electronic subsystem. We investigate the features of the potential in the context of electronic non-adiabatic processes and employ it to study the performance of the classical approximation on nuclear dynamics. We observe that the potential, after the nuclear wave-packet splits at an avoided crossing, develops dynamical steps connecting different regions, along the nuclear coordinate, in which it has the same slope as one or the other adiabatic surface. A detailed analysis of these steps is presented for systems with different non-adiabatic coupling strength. The exact factorization of the electron-nuclear wave-function is at the basis of the decomposition. In particular, the nuclear part is the true nuclear wave-function, solution of a time-dependent Schroedinger euqation and leading to the exact many-body density and current density. As a consequence, the Ehrenfest theorem can be extended to the nuclear subsystem and Hamiltonian, as discussed here with an analytical derivation and numerical results.