Exact Factorization of the Electron–Nuclear Wave Function: Theory and Applications

Federica Agostini, E. K.U. Gross

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

9 Scopus citations


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 languageAmerican English
Title of host publicationQuantum Chemistry and Dynamics of Excited States
Subtitle of host publicationMethods and Applications
Number of pages32
ISBN (Electronic)9781119417774
ISBN (Print)9781119417750
StatePublished - 1 Jan 2020

Bibliographical note

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


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


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