The starting point of essentially all electronic-structure techniques is the Born-Oppenheimer approximation. In the ground state this is usually an excellent approximation. However, when the system is in a highly excited state or strongly driven by external fields, the nuclei experience regions of the potential energy surface with
avoided crossings or conical intersections and one has to face the full Hamiltonian of the complete system of electrons and nuclei. Some of the most fascinating phenomena, such as the process of vision, photovoltaics, as well as decoherence appear in this regime. To tackle this situation, we deduce an exact factorization of the complete electron-nuclear wavefunction into a purely nuclear part and a many-electron wavefunction which
parametrically depends on the nuclear configuration. From this we derive rigorous equations of motion for the nuclear and electronic wavefunctions which lead to a unique definition of exact potential energy surfaces as well as exact geometric phases which are not tied to any adiabatic approximation. For some exactly solvable models, the exact surfaces and geometric phases will be analyzed and compared to their adiabatic counterparts. The equation of motion of the reduced electronic density matrix features a non-unitary time-propagation which turns out to be essential for the ab-initio description of decoherence.
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