Electronic flux densities in vibrating H${}_2$${}^{+}$ in terms of vibronic eigenstates

A theoretical study of the electronic and nuclear flux densities of a vibrating ${{\mathrm{H}}_2}^{+}$ molecular ion is presented. The time-dependent wave function is represented in the basis of vibronic eigenstates which are numerically obtained from the complete nonrelativistic Hamiltonian without the clamped-nuclei approximation. A one-center expansion in terms of $B$-splines and Legendre polynomials is employed to solve the corresponding eigenvalue equation. The electronic and nuclear flux densities are then calculated from the total wave function through their quantum-mechanical definition. Analysis of the flux densities close to the turning points shows that the nuclear wave packet takes longer time (1.4 fs) to change its direction compared to the electronic one (1 fs).

Figure 1

Polar coordinates ($r,\theta$) for the electron. Molecule $\mathit{AB}$ is oriented along the $z$ axis.

Figure 2

Nuclear densities at characteristic times 3.00, 6.45, 11.05, and 27.00 fs. (a) Nuclear probability densities superposed on a plot of the ground-state Born-Oppenheimer potential energy curve; (b) nuclear flux densities.

Figure 3

Electronic probability densities along the internuclear axis ($x=0,y=0$) for the characteristic times 3.00, 6.45, 11.05, and 27.00 fs.

Figure 4

Electronic flux densities: (a) At $t=6.45$ fs for the present results [red (gray) arrows] and results from [2] (black arrows). (b) At $t=11.05$ fs for the present results [red (gray) arrows]; the two solid horizontal lines indicate the electronic wave front located at $z=\pm 1.24a_0$, and the two dashed horizontal lines indicate the nuclear wave front located at $z=\pm 1.18a_0$ [$R_0/2$; see Fig. 2 (b)]. Electronic probability densities are shown as a contour plot.

Figure 5

The $z$ component of the electronic flux density $j_{ez}$ as a function of $z$ for $x=0$, $x=0.22a_0$, $x=0.52a_0$, and $x=0.83a_0$ at $t=11.05$ fs. The arrows represent the direction of the flux, and the vertical dashed lines represent the wave front.

Figure 6

Flux densities at different times: (a) nuclear flux density as a function of $R$; (b) $z$ component of the electronic flux density $j_{ez}$ as a function of $z$ for $x=0$.