Yet it is an approximation, and some of the most fascinating phenomena, such as photovoltaic dynamics, the process of vision, as well as phonon-driven superconductivity occur in the regime where the Born-Oppenheimer approximation breaks down. To tackle such situations one has to face the full Hamiltonian of the complete system of electrons and nuclei. We deduce an exact factorization  of the full electron-nuclear wavefunction into a purely nuclear part and a many-electron wavefunction which parametrically depends on the nuclear configuration and which has the meaning of a conditional probability amplitude. The equations of motion for these wavefunctions lead to a unique definition of exact potential energy surfaces as well as exact geometric phases, both in the time-dependent and in the static case. We discuss a case where the exact Berry phase vanishes although there is a non-trivial Berry phase for the same system in Born-Oppenheimer approximation , implying that in this particular case the Born-Oppenheimer Berry phase is an artifact. In the time-domain, whenever there is a splitting of the nuclear wavepacket in the vicinity of an avoided crossing, the exact time-dependent surface shows a nearly discontinuous step . This makes the classical force on the nuclei jump from one to another adiabatic surface, reminiscent of Tully surface hopping algorithms. Based on this observation, we propose novel mixed-quantum-classical algorithms which provide a rather accurate, much improved (over surface hopping) description of decoherence . We present a multi-component density functional theory  that provides an avenue to make the fully coupled electron-nuclear system tractable in practice. Finally, we apply the concept of exact factorization to a purely electronic wave function, thereby separating, in a formally exact way, fast degrees of freedom (the core electrons) from slow degrees of freedom (electrons that ionize or produce harmonics). This allows us to deduce, in a controlled way, the so-called single-active-electron approximation and systematic improvements thereof .
 A. Abedi, N.T. Maitra, E.K.U. Gross, Phys.
Rev. Lett. 105, 123002 (2010).
 S.K. Min, A. Abedi, K.S. Kim, E.K.U. Gross, Phys. Rev. Lett. 113, 263004 (2014).
 A. Abedi, F. Agostini, Y. Suzuki, E.K.U. Gross, Phys. Rev. Lett. 110, 263001 (2013).
 S.K. Min, F. Agostini, E.K.U. Gross, Phys. Rev. Lett. 115, 073001 (2015).
 R. Requist, E.K.U. Gross, Phys. Rev. Lett. 117, 193001 (2016).
 A. Schild, E.K.U. Gross, Phys. Rev. Lett. (2017, in press).