Non-adiabatic dynamics on a single time-dependent potential energy surface

Some of the most fascinating quantum phenomena, such as the process of vision or phonon-driven superconductivity are examples of non-adiabatic dynamics where more than one Born-Oppenheimer surface is involved. To tackle such situations one has to face the full Hamiltonian of the complete system of electrons and nuclei. We deduce an exact factorization [Abedi, Maitra, Gross, PRL 105, 123002 (2010)] of the full electron-nuclear wavefunction into a purely nuclear part and a many-electron wavefunction which parametrically depends on the nuclear configuration and carries the meaning of a conditional probability amplitude. The equations of motion for these wavefunctions uniquely define the exact potential energy surface and the exact geometric phase, both in the time-dependent and in the stationary regime. 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 [Min, Abedi, Kim, Gross, PRL 113, 263004 (2014)]. 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, moving step [Abedi, Agostini, Suzuki, Gross, PRL 110, 263001 (2013)] between adiabatic surfaces, reminiscent of Tully surface hopping. Based on this observation, we propose novel mixed-quantum-classical algorithms which provide a much improved (over surface hopping) description of decoherence [Min, Agostini, Gross, PRL 115, 073001 (2015)]. This is demonstrated for the laser-induced ring opening of the oxirane molecule [Min, Agostini, Tavernelli, Gross, JPCL 8, 3048 (2017)]. We present a multi-component density functional theory based on the exact factorization that provides an avenue to make the fully coupled electron-nuclear system tractable also for large systems [Requist, Gross, PRL 117, 193001 (2016)].