Exact time-dependent density-functional potentials for strongly correlated tunneling electrons

By propagating the many-body Schrödinger equation, we determine the exact time-dependent Kohn-Sham potential for a system of strongly correlated electrons which undergo field-induced tunneling. Numerous features are entirely absent from the approximations commonly used in time-dependent density-functional theory. The self-interaction correction is strong and time dependent, owing to electron localization, and prominent dynamic spatial potential steps arise from minima in the charge density, as modified by the Coulomb interaction experienced by the partially tunneled electron.

Figure 1

The unperturbed external potential ($V_{\mathrm{ext}}=\alpha x^{10}-\beta x^4$, $t⩽0$, solid red) and its perturbed counterpart (the uniform field $-\varepsilon x$ added, $t>0$, dashed green) (top figure). The interacting charge density (bottom figure) at $t=0$ and at a later time $t=5.3$ a.u. The inset shows a close-up of the density in the tunneling region; at the later time a minimum appears in the density (arrow) as a result of an interference effect (see text).

Figure 2

A zoomed view of the interacting charge density ($n$) in the central region (short dashed blue) and the noninteracting charge density (dotted purple), together with the interacting current density ($j$) (solid red) and the noninteracting current density (dashed green), at $t=5.3$ a.u. The regions of particularly high current-to-charge ratio are indicated by arrows; the modification of this ratio by the Coulomb interaction is evident.

Figure 3

The Hxc potential in the ground state ($t=0$) and at later times $t=3.5$ and $5.3$ a.u. Steps form and grow as a result of minima in the charge density in the regions of finite current density. The distinctive “bump” in the ground-state Hxc potential remains to serve its initial purpose (arrow).

Figure 4

The xc potential at $t=0$ and $5.3$ a.u. In addition to the features noted in $V_{\mathrm{Hxc}}$, the changing strength of the self-interaction correction as the left electron tunnels into the right well is evident.

Figure 5

For $t=5.3$ a.u. the velocity field (dashed green, MB and KS coincide), together with the difference between the Hxc potential at $t=5.3$ a.u. and the ground-state Hxc potential (solid red). The peaks in the velocity field align with the steps in the potential. The largest features correspond to the density most affected by the Coulomb interaction.