Time-dependent formulation of the two-dimensional model of resonant electron collisions with diatomic molecules and interpretation of the vibrational excitation cross sections

A two-dimensional model of the resonant electron-molecule collision processes with one nuclear and one electronic degree of freedom introduced by K. Houfek, T. N. Rescigno, and C. W. McCurdy [Phys. Rev. A 73, 032721 (2006)] is reformulated within the time-dependent framework and solved numerically using the finite-element method with the discrete variable representation basis, the exterior complex scaling method, and the generalized Crank-Nicolson method. On this model we illustrate how the time-dependent calculations can provide deep insight into the origin of oscillatory structures in the vibrational excitation cross sections if one evaluates the cross sections not only at sufficiently large time to obtain the final cross sections, but also at several characteristic times which are given by the evolution of the system. It is shown that all details of these structures, especially asymmetrical peaks, can be understood as quantum interference of several experimentally indistinguishable processes separated in time due to a resonant capture of the electron and the subsequent vibrational motion of the negative molecular ion. Numerical results are presented for the ${\mathrm{N}}_2$-like, NO-like, and ${\mathrm{F}}_2$-like models and compared with ones obtained within the time-independent approach and within the local complex potential approximation.

Figure 1

The full 2D potential energy surface of the ${\mathrm{N}}_2$-like model shown at short distances.

Figure 2

The potential energy curves of the LCP approximation for the ${\mathrm{N}}_2$-like model. The solid black line shows the neutral molecule potential $V_0\left(R\right)$; the dashed blue line marks the resonant energy curve $E_{\mathrm{res}}\left(R\right)$. The shaded area around the resonant energy curve shows the resonance width $\mathrm{\Gamma }\left(R\right)$. The solid red line marks the neutral molecule ground state energy level; the dashed black lines represent several vibrationally excited states.

Figure 3

Snapshots of the evolved wave function for the ${\mathrm{N}}_2$-like model. The evolution time is marked in the left corners. From the top we show the incoming wave packet ($t=50$), the initial reflection to the VE channel ($t=100$), the wave packet trapped behind the potential barrier ($t=150$), and its motion towards larger internuclear distances ($t=300$).

Figure 4

The real (red) and imaginary (purple) part of the wave function of the physical choice of the discrete state ${\varphi }_d(R,r)$ for the ${\mathrm{N}}_2$-like model. Note that orientation of the axes here differs from Fig. 1 for better visibility.

Figure 5

Comparison of the time evolution of the 2D wave function projected onto the discrete state with the LCP wave function for the ${\mathrm{N}}_2$-like model. The time is marked by the color of the curve as indicated in the box on the right side of the panels. The incident wave packets were synchronized by $\mathrm{\Delta }t\doteq 101$. The LCP wave packet was rescaled to match the projection normalization at $t=419$, i.e., multiplied by factor 328 (see Fig. 20 and Sec. 8 for explanation).

Figure 6

The cross sections of the elastic and inelastic processes for the ${\mathrm{N}}_2$-like model at the cutoff time obtained by three methods of the $S$-matrix evaluation and by the LCP approximation are compared with the time-independent approach of the 2D model.

Figure 7

Comparison of the VE $0\to 1$ cross section for the ${\mathrm{N}}_2$-like model with the result of Shandilya et al. [5] and with the experimental result of Allan [21].

Figure 8

The full 2D potential energy surface of the NO-like model shown at short distances.

Figure 9

The potential energy curves $V_0\left(R\right)$ (solid black line) and $E_{\mathrm{res}}\left(R\right)$ (dashed blue line) with the resonance width $\mathrm{\Gamma }\left(R\right)$ marked as the shaded area around $E_{\mathrm{res}}\left(R\right)$ for the NO-like model.

Figure 10

Snapshots of the evolved wave function for the NO-like model. The upper image ($t=100$) shows the initial reflection of the incident wave packet and the second one ($t=200$) capturing of a part of the wave function behind the potential barrier. Note that it is localized in slightly smaller internuclear distance than the incident wave function corresponding to the fact that the probability of capturing is significantly higher for smaller internuclear distances. The third snapshot ($t=630$) shows the trapped wave packet at the maximum of the boomerang motion and the last one ($t=1250$) the penetration of the wave packet back to the VE channels.

Figure 11

Comparison of the time evolution of the 2D wave function projected onto the discrete state with the LCP wave function for the NO-like model. The LCP wave packet was synchronized with the projection by $\mathrm{\Delta }t=176$. For better comparison we have rescaled the LCP wave packet to the projection in time $t=637$ by factor 287 (see Fig. 22 and Sec. 8 for explanation).

Figure 12

The cross sections of the elastic process, vibrational excitation $0\to 1$ and $0\to 8$, and the dissociative attachment for the NO-like model at the cutoff time $t_c=35000$ obtained by three methods of the $S$-matrix evaluation along with the LCP approximation results compared to the time-independent approach of the 2D model.

Figure 13

The full 2D potential energy surface of the ${\mathrm{F}}_2$-like model shown at short distances.

Figure 14

The potential energy curves $V_0\left(R\right)$ (solid black line) and $E_{\mathrm{res}}\left(R\right)$ (dashed blue line) with the resonance width $\mathrm{\Gamma }\left(R\right)$ marked as the shaded area around $E_{\mathrm{res}}\left(R\right)$ for the ${\mathrm{F}}_2$-like model.

Figure 15

Snapshots of the evolved wave function for the ${\mathrm{F}}_2$-like model. The beginning of the evolution is similar to the NO case but there is no reflection of the trapped wave packet at larger internuclear distances. The evolution time is marked in the left corners; the scaling of the figures is marked on the $z$ axis.

Figure 16

The real (red) and imaginary (purple) part of the wave function amplitude of the discrete state ${\varphi }_d(R,r)$ for the ${\mathrm{F}}_2$-like model.

Figure 17

Comparison of the 2D wave function projection onto the discrete state with the LCP wave function for the ${\mathrm{F}}_2$-like model. The LCP wave packet was synchronized with the projection to match in the beginning of the outgoing motion by $\mathrm{\Delta }t=177$. The LCP wave packet was rescaled for better comparison. The evolution time is marked by the color of the curve corresponding to the color bar on the right side of the panels.

Figure 18

The elastic, VE $0\to 1$ and DA cross sections for the ${\mathrm{F}}_2$-like model at the cutoff time $t_c=15000$ obtained by three approaches to the $S$-matrix evaluation.

Figure 19

Comparison of the elastic scattering cross sections obtained by three investigated methods at time $t=900$ for the NO-like model.

Figure 20

The mean internuclear distance of the projections and the LCP wave packet (top panel) and their normalization (bottom panel) as functions of the evolution time. The vertical dashed lines mark the positions of the maxima of the mean internuclear distance. The position of the first maximum at $t=419$ was used to synchronize the evolution within the LCP approximation by $\mathrm{\Delta }t=101$ and also to adjust normalization of the LCP wave packet for better comparison.

Figure 21

The elastic and VE cross sections for two chosen transitions evaluated at significant values of the evolution time. The dashed lines stand for the 2D model, the dotted lines for the LCP approximation. See also Supplemental Material [18] for videos of the cross sections as functions of time.

Figure 22

The mean internuclear distance of the projections and LCP wave packet (top panel) and their normalization (bottom panel) as functions of the evolution time. Both mean internuclear distance and normalization curves were synchronized by the position of the first maximum of the mean internuclear distance at $t=637$, with $\mathrm{\Delta }t=176$.

Figure 23

The same as in Fig. 22 but for times up to 30 000 a.u.

Figure 24

The elastic and VE transition $0\to 1$ cross sections at significant values of the evolution time. All results are for the full 2D NO-like model. See also Supplemental Material [18] for videos of the cross sections as functions of time.

Figure 25

The normalization of the projection of the 2D wave function on the discrete state for the NO-like model and contributions to this normalization of the first four quasibound vibrational states of the molecular negative ion as functions of the evolution time.