Exact Time-Dependent Exchange-Correlation Potential in Electron Scattering Processes

We identify peak and valley structures in the exact exchange-correlation potential of time-dependent density functional theory that are crucial for time-resolved electron scattering in a model one-dimensional system. These structures are completely missed by adiabatic approximations that, consequently, significantly underestimate the scattering probability. A recently proposed nonadiabatic approximation is shown to correctly capture the approach of the electron to the target when the initial Kohn-Sham state is chosen judiciously, and it is more accurate than standard adiabatic functionals but ultimately fails to accurately capture reflection. These results may explain the underestimation of scattering probabilities in some recent studies on molecules and surfaces.

Figure 1

Snapshots of the exact electron density $n(x,t)$ in the $e$-H scattering model system (black-solid line in the upper panel for each time slice). The black line in the lower panel shows the exact time-dependent XC potential $v_{\mathrm{xc}}$ for the initial KS state ${\mathrm{\Phi }}_0^{(1)}$ for each time slice. The results of ALDA (red-solid line) and $v_{\mathrm{xc}}^S$ (blue-solid line) are shown in each panel. The kinetic component of the exact XC potential $v_c^T$ is also shown as green-dotted line in the lower panels.

Figure 2

Number of electrons in the reflection region $N_R$ (left panel) and transmission region $N_T$ (right panel) for the exact (black solid), ALDA (${\mathrm{\Phi }}_0^{(1)}$: red dashed, ${\mathrm{\Phi }}_0^{(2)}$: red solid), $v_{\mathrm{xc}}^{\mathrm{S}}$ (${\mathrm{\Phi }}_0^{(1)}$: blue dashed, ${\mathrm{\Phi }}_0^{(2)}$: blue solid), AEXX (${\mathrm{\Phi }}_0^{(1)}$: equal to $v_{\mathrm{xc}}^{\mathrm{S}}$ (blue dashed), ${\mathrm{\Phi }}_0^{(2)}$: cyan solid) for the two different momenta $p=-1.5$ (upper panels) and $p=-0.3$ (lower panels).

Figure 3

Same as Fig. 1, but for the initial KS state ${\mathrm{\Phi }}_0^{(2)}$.

Figure 4

Snapshots of $v_{\mathrm{xc}}^{\mathrm{AE}}$ (orange dotted) and $v_c^{\mathrm{AE},T}$ (dark green). Exact electron density is also shown as black-dotted line.