Qualitative failure of a multiconfiguration method in prolate spheroidal coordinates in calculating dissociative photoionization of ${\mathrm{H}}_2{}^{+}$

A formulation of a multiconfiguration time-dependent Hartree-Fock (MCTDHF) method with nuclear motion is tested by application to a three-body breakup problem, the dissociative photoionization cross section of the ${\mathrm{H}}_2^{+}$ ion. The representation of the wave function in terms of a set of Slater determinants used for all nuclear geometries, with a prescribed parametric dependence upon the nuclear geometry such that the cusps follow the nuclei, times a complete basis expansion in the nuclear degrees of freedom shows promise as a method for treating nonadiabatic electronic and nuclear motion in molecules. However, the method used here for diatomics, in which the parametric dependence is prescribed through the choice of prolate spheroidal coordinates, produces qualitatively incorrect steplike behavior in the calculated cross section near onset. Modifications to the prolate spheroidal coordinate system that would improve this nonadiabatic diatomic MCTDHF representation are proposed.

Figure 1

Cross sections for photoionization of ${\mathrm{H}}_2{}^{+}$, including nonadiabatic nuclear motion in the bond length, calculated with one to four orbitals, compared to the Born-Oppenheimer result and to the Chase approximation. These are the results of four calculations, using two different laser pulses, one used to calculate the cross section up to 80 eV and one for the onset region, for both parallel, left, and perpendicular, right, polarization.

Figure 2

Schematic of wave-function density, $|\mathrm{\Psi }(\vec{r},R){|}^2$, integrated over angular coordinates, for an ionized wave packet created by a short pulse, in spherical polar coordinates, left, and prolate spheroidal coordinates, middle. The hypothetical closest approximation to this wave packet using two orbitals in prolate coordinates is on the right.

Figure 3

Initial natural orbitals (real valued) $\varphi (\vec{r};R)$, evaluated at $R=2.0a_0$, $y=0$, for two-orbital calculation.

Figure 4

Natural orbitals $\varphi (\vec{r};R)$ for two-orbital propagation during the pulse, as the ionized flux moves outward, at, from top to bottom, $t=$ 0, 0.5, and 1.0 fs after the start of the pulse, evaluated at $R=2.0a_0$, $y$=0. In each panel, the left image is the real part; the right, translated by 80 bohrs, is the imaginary. The left three panels are the first natural orbital (occupancy approximately 96%) and the right are the second (4%).

Figure 5

Natural orbitals $\varphi (\vec{r};2.0)$ for two-orbital propagation after the pulse, at $t=0$, 1.75, and 3.75 fs after the end of the pulse, as the wave function is ringing down, plotted in the same way as Fig. 4. At $R=2.0a_0$, the wave function is absorbed via complex coordinate scaling starting at approximately $r=80a_0$.