Exact factorization of a many-body wavefunction beyond the electron-nuclear problem

Abstract

This Review is devoted to the presentation of the exact factorization as a framework employed to study a variety of quantum-mechanical many-body problems. Since its original formulation in the 70s, the main applications of the exact factorization were directed towards understanding the properties of multi-component systems, e.g. electron-nuclear systems. Especially in the electron-nuclear case, the exact factorization is viewed as an exactification of the Born-Oppenheimer approximation, thus it was, and still is, largely employed in nonadiabatic dynamics. Nonetheless, as early as the 80s, the formalism was employed to study many-electron interacting systems and, quite recently, i.e. less than a decade ago, it was extended to study the behavior of molecules in the context in cavity quantum electrodynamics. These formulations, perhaps less popular than the electron-nuclear formulation, have attracted a lot of attention over the years. Therefore, we review here the exact electron-only factorization and the exact photon-electron-nuclear factorization.

Figures (14)

• Figure 1: Demonstration of the step character of the potential $v_{\text{s},N-1}$ for Kr (a) and Cd (b). Reproduced from Gritsenko, O., van Leeuwen, R., & Baerends, E. J. (1994). The Journal of chemical physics, 101(10), 8955-8963, with the permission of AIP Publishing.
• Figure 2: Response potential for the two-electron singlet system of a model diatomic molecule shown at various internuclear distances $R$ (a.u.), labeled on the corresponding curves. The position of the right nucleus at $R/2$ is indicated by a dot. Adapted from Giarrusso, S., Neugarten, R., Baerends, E. J., & Giesbertz, K. J. (2022). Journal of chemical theory and computation, 18(8), 4762-4773, with the permission of the American Chemical Society.
• Figure 3: Approximate Hxc potential obtained from an approximate conditional density (blue) compared with the exact one (red) for three different internuclear distances: $R = 3\, a_0$ (top panel), $7\, a_0$ (middle panel), and $11\, a_0$ (bottom panel). In the most stretched geometries, the approximate potential accurately reproduces the step structure. Adapted from Giarrusso, S., & Agostini, F. (2025). The Journal of Chemical Physics, 162(9), licensed under CC BY.
• Figure 4: Electronic vector potential for two non-interacting fermions in a triplet state in Cartesian coordinates $x, \, y, \, z$ for nuclear charge $Z=1$. Color coding reflects the magnitude of the vector fields at each point. Adapted from Giarrusso, S., Gori-Giorgi, P., & Agostini, F. (2024). ChemPhysChem, 25(18), e202400127, licensed under CC BY.
• Figure 5: Pictorial representation of a molecule in an optical cavity with frequency $\omega_c$. The photon-electron-nuclear system is described by the many-body wavefunciton $\Psi(\bm{r},\bm{R},\bm{q},t)$.