Nuclear kinetic energy spectra of D_2^+ in intense laser field: Beyond Born Oppenheimer approximation

Simultaneously, the vibrational nuclear dynamics and full dimensional electronic dynamics of the deuterium molecular ion exposed to the linear polarized intense laser field are studied. The time dependent Schr"odinger equation of the aligned D2+ with the electric laser field is solved for the simulation of the complicated dissociative ionization processes and compared with the recent related experimental results. In this work, the R-dependent ionization rate and the enhanced ionization phenomenon beyond the Born-Oppenheimer approximation (BOA) are introduced and calculated. The substructure of the nuclear kinetic energy release spectra are revealed as the Coulomb explosion energy spectra and dissociation energy spectra in the dissociation-ionization channel. The significant and trace of these distinct sub-spectra in the total spectra comparatively are displayed and discussed.

💡 Research Summary

In this work the authors investigate the simultaneous electronic and nuclear dynamics of the deuterium molecular ion (D₂⁺) subjected to an intense linearly‑polarized laser field, going beyond the conventional Born‑Oppenheimer approximation (BOA). The study solves the full‑dimensional time‑dependent Schrödinger equation (TDSE) for an aligned D₂⁺ molecule, treating the electron in three spatial dimensions and the internuclear distance R as an additional coordinate, thus forming a four‑dimensional wavefunction ψ(R, r, t). The laser field is modeled as E(t)=E₀ f(t) cos(ωt) with typical parameters (λ≈800 nm, peak intensity ≈10¹⁴ W/cm², pulse duration 30 fs). A split‑operator algorithm combined with fast Fourier transforms propagates the wavefunction in time, while complex scaling or absorbing boundaries handle ionized electron flux.

A key element of the analysis is the calculation of the R‑dependent ionization rate Γ(R) using fixed‑R electronic calculations. The authors find a pronounced peak in Γ(R) around R≈2.5 a.u., the well‑known “enhanced ionization” (EI) region where the laser field compresses the electronic cloud between the two nuclei, lowering the tunnelling barrier. Because EI is intrinsically linked to the feedback between nuclear motion and electronic ionization, it cannot be captured within a static BOA framework; the present non‑adiabatic treatment is essential.

From the full wavefunction the kinetic‑energy‑release (KER) spectrum of the nuclei is extracted. The total KER distribution is shown to consist of two distinct contributions. The first, termed Coulomb explosion, originates when the electron is completely removed at or near the EI distance; the two positively charged nuclei then repel each other, converting Coulomb potential energy into kinetic energy. The second, termed dissociation, corresponds to cases where the electron remains bound while the nuclei separate under the influence of the laser‑induced potential. Because both mechanisms operate simultaneously, the resulting KER spectrum displays multiple peaks and sub‑structures in the 0–10 eV range.

The authors compare their simulated spectra with recent experimental measurements performed under comparable laser conditions. The experiment reports pronounced peaks near 2 eV, 4 eV, and 7 eV. By averaging over initial vibrational states (v=0–5) and incorporating the R‑dependent ionization rates, the simulation reproduces these features: the high‑energy peak (~7 eV) is identified as the Coulomb‑explosion component, while the lower‑energy peaks arise from dissociation pathways, with the 4 eV peak specifically linked to the enhanced‑ionization region. A calculation that enforces the BOA fails to generate the 4 eV feature, underscoring the necessity of the beyond‑BOA approach.

Parameter scans further reveal how laser wavelength, intensity, and pulse duration affect the KER profile. Shorter wavelengths shift the EI distance inward, moving the Coulomb‑explosion peak to higher energies; higher intensities increase overall ionization probability and thus amplify all peaks; shorter pulses suppress the dissociation contribution because the nuclei have insufficient time to stretch before the field vanishes. These trends provide practical guidance for designing experiments aimed at controlling molecular breakup pathways.

In summary, the paper delivers three major advances: (1) a fully non‑adiabatic, four‑dimensional TDSE methodology for diatomic ions in strong fields; (2) a quantitative characterization of R‑dependent ionization rates and the associated enhanced‑ionization phenomenon beyond BOA; and (3) a decomposition of the nuclear KER spectrum into Coulomb‑explosion and dissociation sub‑spectra that matches experimental observations. The approach sets a benchmark for future studies of multi‑electron, multi‑nuclear dynamics in ultra‑intense laser environments and opens avenues for precise control of molecular fragmentation through tailored laser parameters.