Attosecond photoelectron microscopy of H₂⁺

We present a numerical study of the ultrafast ionization dynamics of H₂⁺ exposed to attosecond extreme ultraviolet (xuv) pulses that goes beyond the Born-Openheimer approximation. The four-dimensional, time-dependent Schrödinger equation was solved using a generalization of the finite-element discrete-variable-representation/real-space–product technique used in our previous calculations to include the dynamical motion of the nuclei. This has enabled us to expose the target to any polarized light at arbitrary angles with respect to the molecular axis. Calculations have been performed at different angles and photon energies (ℏω=50 eV up to 630 eV) to investigate the energy and orientation dependence of the photoionization probability. A strong orientation dependence of the photoionization probability of H₂⁺ was found at a photon energy of ℏω=50 eV. At this energy, we found that the ionization probability is three times larger in the perpendicular polarization than in the parallel case. These observations are explained by the different geometric “cross sections” seen by the photoejected electron as it leaves the molecule. This ionization anisotropy vanishes at the higher-photon energy of ℏω≥170 eV. When these higher-energy xuv pulses are polarized perpendicular to the internuclear axis, a “double-slit-like” interference pattern is observed. However, we find that the diffraction angle only approaches the classical formula ϕ_n=sin⁻¹(nλ_e/R₀), where n is the diffraction order, λ_e is the released electron wavelength, and R₀ is the internuclear distance, when nλ_e becomes less than 65% of R₀. These results illustrate the possibility of employing attosecond pulses to perform photoelectron microscopy of molecules.