Retrieval of the nuclear motion in a molecule from photoelectron momentum distributions using non-Born-Oppenheimer quantum dynamics and deep learning

By using a neural network that takes momentum distributions of photoelectrons produced in strong-field ionization as input, we retrieve the time-dependent bond length of a dissociating one-dimensional H$_{2}^{+}$ molecule. The photoelectron momentum distributions are calculated from the direct numerical solution of the non-Born-Oppenheimer time-dependent Schrödinger equation. We simulate two setups: first, molecules prepared in the first excited electronic state, second, a pump-probe scheme starting from the ground state. We show that in both schemes a neural network trained on momentum distributions calculated for frozen nuclei retrieves the time-dependent bond length with an absolute error of $0.2$-$0.4$ a.u. We find that a neural network, when applied to photoelectron momentum distributions obtained within the pump-probe scheme, can be used for the retrieval of the equilibrium internuclear distance and ground-state population. This opens new perspectives for extracting electronic properties of molecules from electron momentum distributions using deep learning.

💡 Research Summary

In this work the authors combine fully quantum non‑Born‑Oppenheimer (NBO) simulations with deep learning to retrieve the time‑dependent internuclear distance of a dissociating H₂⁺ ion from strong‑field photo‑electron momentum distributions (PMDs). The study focuses on a one‑dimensional model of H₂⁺ in which both the electron coordinate (x) and the internuclear distance (R) are treated quantum mechanically. The time‑dependent Schrödinger equation (TDSE) is solved directly using the Feit‑Fleck‑Steiger split‑operator method on a large grid (x ∈