Charge transfer in time-dependent density-functional theory: Insights from the asymmetric Hubbard dimer

We show that propagation with the best possible adiabatic approximation in time-dependent density-functional theory fails to properly transfer charge in an asymmetric two-site Hubbard model when beginning in the ground state. The approximation is adiabatic but exact otherwise, constructed from the exact ground-state exchange-correlation functional that we compute via constrained search. The model shares the essential features of charge-transfer dynamics in a real-space long-range molecule, so the results imply that the best possible adiabatic approximation, despite being able to capture nonlocal ground-state step features relevant to dissociation and charge-transfer excitations, cannot capture fully time-resolved charge-transfer dynamics out of the ground state.

Figure 1

Exact $E_{\mathrm{HXC}}\left[\Delta n\right]$ (red dashed), g.s. potential $\Delta v_{\mathrm{HXC}}^{\mathrm{g}.\mathrm{s}.}\left[\Delta n\right]$ (black solid), and scaled g.s. kernel $\Delta f_{\mathrm{HXC}}^{\mathrm{g}.\mathrm{s}.}\left[\Delta n\right]/25$ (blue dotted) for $T/U=0.05$. The inset shows the correlation potential $\Delta v_{\mathrm{C}}^{\mathrm{g}.\mathrm{s}.}\left[\Delta n\right]$ for $T/U=0.05$ (black solid), $T/U=0.1$ (pink dashed), and the HX potential $\Delta v_{\mathrm{HX}}^{\mathrm{g}.\mathrm{s}.}\left[\Delta n\right]$ (independent of $T$, orange dotted). All functionals are in units of $U$.

Figure 2

Exact dipole $\Delta n$ (black solid), AE dipole $\Delta n_{sc}^{\mathrm{AE}}$ (red dashed), and the AEXX dipole $\Delta n_{sc}^{\mathrm{AEXX}}$ (pink dotted). Left panel is for cs-cs CT. Right panel is for os-os CT. Time is in units of $1/U$.

Figure 3

Left panel: Exact dipole $\Delta n$ (black solid) and AE dipole $\Delta n_{sc}^{\mathrm{AE}}$ (red dashed) for different values of the static potential difference $\Delta v^0$ and resonant driving. Right panel: Same quantities but for fixed value of the static potential $\Delta v^0=-2.0U$ and different detunings ${\omega }_{\text{det}}=\omega -{\omega }_{\mathrm{CT}}$ of the laser frequency $\omega$ with respect to the interacting resonance ${\omega }_{\mathrm{CT}}=1.0083U$. In all cases the amplitude of the laser is $0.1U$. Time is in units of $1/U$.

Figure 4

(Upper panel) Exact KS potential $\Delta v_{\mathrm{S}}$ [black solid; first term on right-hand side of Eq. (2)] and the field $\mathcal{E}\left(t\right)$ (pink dashed). (Middle panel) Exact HXC potential Eq. (2) (black solid), the AE HXC potential $\Delta v_{\mathrm{HXC}}^{\mathrm{AE}}\left[\Delta n\right]$ (blue dashed), and the AE HXC potential $\Delta v_{\mathrm{HXC}}^{\mathrm{AE}}\left[\Delta n_{sc}\right]$ (red dotted) with self-consistent $\Delta n_{sc}=\Delta n_{sc}^{\mathrm{AE}}$. (Lower panel) Correlation potentials. The left panels show cs-cs CT. The right panels show os-os CT. Time is in units of $1/U$.