Initial-state dependence of coupled electronic and nuclear fluxes in molecules

We demonstrate that coupled electronic and nuclear fluxes in molecules can strongly depend on the initial state preparation. Starting the dynamics of an aligned ${\mathrm{D}}_2{}^{+}$ molecule at two different initial conditions, the inner and the outer turning points, we observe qualitatively different oscillation patterns of the nuclear fluxes developing after 30 fs. This corresponds to different orders of magnitude bridged by the time evolution of the nuclear dispersion. Moreover, there are attosecond time intervals within which the electronic fluxes do not adapt to the nuclei motion depending on the initial state. These results are inferred from two different approaches for the numerical flux simulation, which are both in good agreement.

Figure 1

(a) The potential energy surface $V(R)$ of ${\mathrm{D}}_2{}^{+}$. The outer (1) and the inner (2) regions denote the initial locations of the nuclear wave packets centered at $R_1$ and $R_2$, respectively. These wave packets have been promoted to ${\mathrm{D}}_2{}^{+}$ from the ground state of ${\mathrm{D}}_2$. The dot-dashed vertical lines indicate the location of our observer $R_{\mathrm{obs}}$. (b) The force $K=-\mathit{dV}/\mathit{dR}$ acting on the nuclear wave packet. The equilibrium internuclear distance $R_{\mathrm{eq}}$ is indicated by a dashed vertical line.

Figure 2

Visualization of the initial states as defined in Fig. 1. The upper panel represents the inner initial state and the lower panel the outer one. The electronic density, with its associated color code, surrounds the nuclear densities in red (dark gray). The space inside the two parallel planes of the observer $A_{\mathrm{obs}}$, in yellow (light gray), corresponds to the volume of the observer $V_{\mathrm{obs}}$. The two planes are located at $z_{\mathrm{obs}}=\pm R_{\mathrm{obs}}/2$, where $R_{\mathrm{obs}}$ is the distance separating the two planes. This visualization has been created using the academic system ZIBAMIRA, a superset of its commercial version AMIRA [15].

Figure 3

Dynamics for the outer initial condition centered in $R_1=3.125a_0$. The underlying simulations solve the full molecular Schrödinger equation [red (gray)] or use the BO approximation (black). Nuclear quantities are plotted by solid lines, electronic ones by dashed lines. (a) The mean bond length. Additionally, the observer location at $R_{\mathrm{obs}}=2.5a_0$ is indicated as a dashed horizontal line. The lower two panels show the evolution of the electronic and nuclear fluxes (b) and the yields (c). The dynamics within the regions marked by A and B are more thoroughly discussed in subsection 4b.

Figure 4

Dynamics for the inner initial condition centered in $R_2=1.5a_0$. The displayed quantities as well as their line encoding are the same as in Fig. 3.

Figure 5

Robustness of the electronic and nuclear flux with respect to the observer position. The nuclear flux evolution for the outer and the inner initial state are shown in the upper and the lower panels, respectively. The three different locations of the observer are $R_{\mathrm{obs}}=2.3a_0$ (dashed), $2.4a_0$ [red (gray)], and $2.5a_0$ (black). Both panels also plot the electronic flux (dot-dashed), which is insensitive to the considered variations of the observer plane.

Figure 6

Electronic and nuclear dispersions as functions of time: the nuclear one $\Delta R(t)$ (solid), the electronic one in $z$ direction $\Delta z(t)$ (dashed) and $r$ direction $\Delta r(t)$ (dotted). The dynamics for the outer intial condition (1) are in black, while those for the inner one (2) are in [red (gray)] colors.

Figure 7

Enlargements of regions A and B of Fig. 3b for the nuclear flux (solid) and the electronic one (dashed). The electron and nuclei start moving synchronously (A-Outer). At later time (B-Outer), the electron returns first and after 840 as, the nuclei follow.

Figure 8

Enlargements of regions A and B of Fig. 4b for the nuclear flux (solid) and the electronic one (dashed). Until 3.5 fs, the electron and nuclei move in opposite directions (A-Inner). At later time (B-Inner), the nuclei return and the electron follows with a delay of 1.04 fs.