Role of nuclei movement in the tunneling ionization of hydrogen molecular ions beyond the dipole approximation

$H_2^{+}$ exposed to strong mid-infrared laser pulses undergoes the tunneling ionization, followed by the rescattering of the photoelectron with the parent ion. The two nuclei repel each other during the tunneling and rescattering, which modifies the Coulomb potential experienced by the rescattering electron. We numerically study the role of the nuclear movement in the tunneling ionization of hydrogen molecular ions. The classical trajectory Monte-Carlo simulations give intuitive explanations of photoelectron momentum distributions driven by laser pulses having different wavelengths and different intensities. The competition of the Coulomb attraction and the Lorentz push determines the photoelectron momentum distribution along the laser propagation direction. The nuclear movement modifies the Coulomb focusing, and thus shifts the peaks of the photoelectron momentum distribution along the laser propagation direction. (paper)