Effect of moving nuclei in multiphoton ionization of the H$_2^{\,+}$ ion

We propose an accurate {\it ab initio} numerical method to depict the dynamics of the nuclear fragments and the entangled motion of the nuclei and the electron in the laser-driven H$_2^{\,+}$ ion. Building on recent work~[1], we solve the time-dependent Schr\"odinger equation in prolate spheroidal coordinates and extract the angle-differential cross section for the photo\-electron as well as the kinetic energy release spectra of the nuclei. Assuming that the nuclei are frozen in their lowest rotational state, the nuclear coordinate in the wave function is discretized through a finite-element discrete-variable representation (FE-DVR), which is coupled to other FE-DVRs for the electron coordinate. The present procedure in full dimensionality allows us to go beyond the popular fixed-nuclei approximation, the two-channel approximation, and reduced-dimensional models. As a specific example, we discuss the effect of the nuclear motion in the H$_2^{\,+}$ ion followed by two- and three-photon absorption, both in the parallel and perpendicular geometries.\\[4pt] [1] X.~Guan, E.~Secor, K.~Bartschat, and B.~I.~Schneider, Phys.~Rev.~A~{\bf 84} (2011) 032420.