Hydrodynamic perspective on memory in time-dependent density-functional theory

The adiabatic approximation of time-dependent density-functional theory is studied in the context of nonlinear excitations of two-electron singlet systems. We compare the exact time evolution of these systems to the adiabatically exact one obtained from time-dependent Kohn-Sham calculations relying on the exact ground-state exchange-correlation potential. Thus, we can show under which conditions the adiabatic approximation breaks down and memory effects become important. The hydrodynamic formulation of quantum mechanics allows us to interpret these results and relate them to dissipative effects in the Kohn-Sham system. We show how the breakdown of the adiabatic approximation can be inferred from the rate of change of the ground-state noninteracting kinetic energy.

Figure 1

Time evolution of $X\left(t\right)$ for the 1D two-electron Hooke’s atom with $v_{\text{ext}}\left(z\right)=\left(k/2\right)z^2+E_{0}z\text{sin}\left({\omega }_{\text{f}}t\right)$, where $k=0.1\text{a}\text{.u}\text{.}$, $E_0=0.447\text{a}\text{.u}\text{.}$, and ${\omega }_{\text{f}}=1.870\text{a}\text{.u}\text{.}$ The TDSE (dashed) and AE-TDKSE (dotted-dashed) curves both lie on top of the dotted one given by Eq. (12). For comparison, we also show $E\left(t\right)=E_0\text{sin}\left({\omega }_{\text{f}}t\right)$ (thin solid line).

Figure 2

Time evolution of $E_{\text{h}}$ for A6-Hooke I system calculated with TDSE (solid line) and AE-TDKSE (dotted line).

Figure 3

Time evolution of $T_{\text{s},0}$ for A6-Hooke I system calculated with TDSE (solid line) and AE-TDKSE (dotted line).

Figure 4

Snapshots of exact density $n$ and velocity $u_{\text{s}}$ during A6-Hooke I process. According to Eq. (13) the velocity increases as the density drops (e.g., around $z=4\text{a}\text{.u}\text{.}$). The resulting velocity gradients in regions of very low density are discussed further below.

Figure 5

Time evolution of $E_{\text{h}}$ for A6-Hooke II system calculated with TDSE (solid line) and AE-TDKSE (dotted line).

Figure 6

Time evolution of $T_{\text{s},0}$ for A6-Hooke II system calculated with TDSE (solid line) and AE-TDKSE (dotted line).

Figure 7

Comparison of densities and velocities from TDSE (solid line) and AE-TDKSE (dotted line) schemes at $t=3.36\text{a}\text{.u}\text{.}$ for A6-Hooke II system.

Figure 8

Comparison of densities and velocities from TDSE (solid line) and AE-TDKSE (dotted line) schemes at $t=4.20\text{a}\text{.u}\text{.}$ for A6-Hooke II system.

Figure 9

Comparison of densities and velocities from TDSE (solid line) and AE-TDKSE (dotted line) schemes at $t=5.46\text{a}\text{.u}\text{.}$ for A6-Hooke II system.

Figure 10

Exact evolution of ${\dot{T}}_{\text{s},0}$ for A6-Hooke systems I (without memory, solid line) and II (with memory, dashed line). The dotted-dashed line represents the memory criterion according to Table and formula (25).

Figure 11

Exact evolution of ${\dot{T}}_{\text{s},0}$ for A6-Hooke processes described in Table (without memory: solid lines; with memory: dashed line). The dotted-dashed line represents the memory criterion according to Table and formula (25).

Figure 12

Exact evolution of ${\dot{T}}_{\text{s},0}$ for A4-Hooke systems I (without memory, solid line), II (with memory, dotted line), and III (with memory, dashed line). The dotted-dashed line represents the memory criterion according to Table and formula (25).

Figure 13

Exact evolution of ${\dot{T}}_{\text{s},0}$ for helium systems ramp (without memory, solid line), pulse with $I=7\times {10}^{14}\text{W}/{\text{cm}}^2$ (without memory, dotted line), and oscillating nucleus (with memory, dashed line) as investigated in Ref. [11]. The dotted-dashed line represents the memory criterion according to Table and formula (25). For the pulse ${\dot{T}}_{\text{s},0}$ is reaching finite amplitudes at later times but not exceeding 40% of ${\dot{T}}_{\text{s},0}^{\text{crit}}$.