## Abstract

In this Chapter we review the exact factorization of the electron-nuclear wave function. The molecular wave function, solution of a time-dependent Schröodinger equation, is factored into a nuclear wave function and an electronic wave function with parametric dependence on nuclear configuration. This factorization resembles the (approximate) adiabatic product of a single Born-Oppenheimer state and a time-dependent nuclear wave packet, but it introduces a fundamental difference: both terms of the product are explicitly time-dependent. Such feature introduces new concepts of time-dependent vector potential and time-dependent potential energy surface that allow for the treatment of nonadiabatic dynamics, thus of dynamics beyond the Born-Oppenheimer approximation. The theoretical framework of the exact factorization is presented, also in connection to the more standard Born-Huang (still exact) representation of the molecular wave function. A trajectory-based approach to nonadiabatic dynamics is derived fromthe exact factorization. A discussion on the connection between the molecular Berry phase and the corresponding quantity arising from the exact factorization is briefly discussed.

Original language | American English |
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Title of host publication | Quantum Chemistry and Dynamics of Excited States |

Subtitle of host publication | Methods and Applications |

Publisher | wiley |

Pages | 531-562 |

Number of pages | 32 |

ISBN (Electronic) | 9781119417774 |

ISBN (Print) | 9781119417750 |

DOIs | |

State | Published - 1 Jan 2020 |

### Bibliographical note

Publisher Copyright:© 2021 John Wiley & Sons Ltd. All rights reserved.

## Keywords

- Born-Oppenheimer framework
- Electron-nuclear wave function
- Exact factorization
- Molecular berry phase
- Time-dependent molecular