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

TL;DR

The paper tackles the challenge of extracting real-time nuclear motion and electronic-state information from photoelectron momentum distributions in strong-field molecular ionization, using a non-Born-Oppenheimer framework for a 1D $H_{2}^{+}$ model. It solves the full non-BO time-dependent Schrödinger equation to generate PMDs under two excitation schemes—instantaneous excitation to the first excited state and a pump-probe setup—and trains neural networks (fully-connected and CNN) on PMDs, including data computed with frozen nuclei. The results show that networks trained on fixed-nuclei PMDs can retrieve the time-dependent bond length with absolute errors around $0.2$-$0.4$ a.u., and, in the pump-probe case, can also reconstruct ground-state populations when PMDs are treated as incoherent sums of contributions from different electronic states. When networks are applied to moving-nuclei PMDs, performance degrades unless the training data account for nuclear motion; by combining PMDs via weighted incoherent sums and tuning the pump-probe intensity, the authors demonstrate accurate retrieval of $R(t)$ (MAE ≈ $0.19$ a.u. for the excited-state component) and successful estimation of initial bond length and state populations. Overall, the work demonstrates a practical data-driven route to time-resolved molecular imaging and electronic-property retrieval from PMDs, extending beyond Born-Oppenheimer approximations and highlighting the potential for real-time molecular dynamics probes guided by deep learning.

Abstract

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.

Figures (4)

• Figure 1: The architecture of the (a), (b) FCCNs and (c) CNN used for the retrieval of the bond length $R$. Both neural networks take 1D electron momentum distributions normalized to their maximum value as input; see text.
• Figure 2: Electron momentum distributions for ionization of the 1D H$_2^{+}$ molecule calculated from the direct numerical solution of the TDSE for the internuclear distance $R=9.26$ a.u. The molecule is ionized by a laser pulse with duration $n_{p}=2$ cycles, peak intensity $2.5\times 10^{14}$ W/cm$^2$, and wavelength $800$ nm. (a) Momentum distribution obtained from the moving-nuclei TDSE for instantaneous excitation of the molecule and ionization at $\langle R \rangle=9.26$ a.u. (b) Distribution calculated within the pump-probe scheme. The wavelength of the pump pulse is $148$ nm ($\omega=0.368$ a.u.), the peak laser intensity is $8.4\times10^{12}$ W/cm$^2$, and duration of this pulse is $n_p=3$ optical cycles. (c) Distribution obtained from the solution of the fixed-nuclei TDSE (\ref{['tdse_1d']}) for ionization of the ground state. (d) Distribution calculated from Eq. (\ref{['tdse_1d']}) for ionization of the first excited state. The distributions are normalized to their maximum value.
• Figure 3: Plots of predicted vs true internuclear distances [(a),(c)] and laser intensities [(b),(d)] illustrating the performance of neural networks for the case of instantaneous excitation. (a) and (b) correspond to an FCCN trained on a set of distributions calculated from the TDSE (\ref{['tdse_main']}) with moving nuclei. (c) and (d) show the performance of an FCCN trained on a set of PMDs with fixed nuclei, i.e., obtained from the solution of Eq. (\ref{['tdse_1d']}), for ionization from the first excited state. The dashed curves in (a) and (c) show the time-dependent expectation value of the bond length.
• Figure 4: Predictions of neural networks for the internucelar distance during dissociation (blue points) at different time delays compared to the time-dependent expectation value of the internuclear distance (red dashed curves) obtained from the solution of the TDSE (\ref{['tdse_main']}) for the pump-probe scheme. (a) Results from the FCCN trained on momentum distributions calculated from Eq. (\ref{['set_pmd']}) for fixed $R_0=2.59$ a.u. The pump pulse intensity is $3.29\times10^{10}$ W/cm$^2$ . (b) Results from the FCCN trained on the set of distributions (\ref{['set_pmd']}) with $R_{0,i}$ randomly distributed in the range $\left[1.0, 3.0\right]$ a.u. The PMDs of the test set are again calculated for the intensity of the pump pulse equal to $3.29\times10^{10}$ W/cm$^2$.