Nonadiabatic electron dynamics in time-dependent density-functional theory

Time-dependent density-functional theory (TDDFT) treats dynamical exchange and correlation (xc) via a single-particle potential, $V_{\mathrm{xc}}(\mathbf{r},t)$, defined as a nonlocal functional of the density $n({\mathbf{r}}^{\prime },t^{\prime })$. The popular adiabatic local-density approximation (ALDA) for $V_{\mathrm{xc}}(\mathbf{r},t)$ uses only densities at the same space-time point $(\mathbf{r},t)$. To go beyond the ALDA, two local approximations have been proposed based on quantum hydrodynamics and elasticity theory: (a) using the current as the basic variable (C-TDDFT) [G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett. 79, 4847 (1997)], (b) working in a comoving Lagrangian reference frame (L-TDDFT) [I. V. Tokatly, Phys. Rev. B 71, 165105 (2005)]. In this paper we illustrate, compare, and analyze both nonadiabatic theories for simple time-dependent model densities in the linear and nonlinear regime, for a broad range of time and frequency scales. C- and L-TDDFT are identical in certain limits, but, in general, exhibit qualitative and quantitative differences in their respective treatment of elastic and dissipative electron dynamics. In situations where the electronic density rapidly undergoes large deformations, it is found that nonadiabatic effects can become significant, causing the ALDA to break down.

Figure 1

Scaled memory kernel $Y(n,t-t^{\prime })$ for $r_s=1,2,3,4,5$, using the GK (full lines) and QV (dashed lines) parametrizations. Inset: short-time limit $Y(n,0)$ [Eq. (22)] versus $r_s$.

Figure 2

Snapshots of density $n(x,t)$, in units of $N∕L$, velocity $v(x,t)$, in units of $L\omega$, and deformation $\overline{g}(x,t)$ for the sloshing mode in the laboratory frame, taken at times $t=0,T∕4,T∕2,3T∕4$.

Figure 3

The same as Fig. 2, for the breathing mode.

Figure 4

(Color online) Snapshots of ALDA xc potential fluctuations and xc memory contributions during one cycle of the sloshing mode (Fig. 2), with amplitude $A=0.005$. Blue (dark gray) solid lines: ${\tilde{V}}_{\mathrm{xc}}^{\mathrm{ALDA}}$, scaled by 0.1 (independent of frequency). Red (medium gray) solid lines: $V_{\mathrm{xc}}^M$, in GK parametrization, for low $\left(L\right)$ frequency, ${\omega }_L=0.001{\overline{\omega }}_p$, crossover $\left(C\right)$ frequency, ${\omega }_C=1.7{\overline{\omega }}_p$, and high $\left(H\right)$ frequency, ${\omega }_H=1000{\overline{\omega }}_p$. Green (light gray) dashed lines: $V_{\mathrm{xc}}^M$ at 0.1, 0.5, 2, 10 times ${\omega }_C$.

Figure 5

(Color online) The same as Fig. 4, but $V_{\mathrm{xc}}^M$ calculated in QV parametrization. The crossover frequency is ${\omega }_C=2.15{\overline{\omega }}_p$.

Figure 6

(Color online) The same as Fig. 4, but for the breathing mode (Fig. 3). ${\tilde{V}}_{\mathrm{xc}}^{\mathrm{ALDA}}$ is not scaled. The crossover frequency is ${\omega }_C=2.22{\overline{\omega }}_p$.

Figure 7

(Color online) The same as Fig. 6, but $V_{\mathrm{xc}}^M$ calculated in QV parametrization. The crossover frequency is ${\omega }_C=2.65{\overline{\omega }}_p$.

Figure 8

Power $\mathcal{P}\left(t\right)$ [Eq. (58)] calculated using GK, associated with the sloshing and the breathing mode, during one cycle of the charge-density oscillation $(T=2\pi ∕\omega )$, for different frequencies. $L$, $C$, $H$ indicate the low-frequency (purely dissipative in GK), crossover, and high-frequency regime (purely elastic).

Figure 9

Full symbols: net rate of power absorption, $-\overline{\mathcal{P}}∕\omega A^2$, during one cycle of the sloshing and breathing modes, versus frequency (in GK). Full line: $-\gamma \mathfrak{I}{f}_{\mathrm{xc}}^L\left(\omega \right)$, for $r_s=1.41$ (sloshing) and 1.18 (breathing). Open symbols: Phase lag (in units of $\pi$) between $\mathcal{P}\left(t\right)$ and ALDA. The crossover frequency is defined by a maximum in power absorption and a phase lag $3\pi ∕4$. The dashed lines are a guide to the eye.

Figure 10

The same as Fig. 9, but in QV parametrization. Both the low- and high-frequency limits are purely elastic.

Figure 11

Power $\mathcal{P}\left(t\right)$ [Eq. (58)] calculated using GK, over one cycle of the sloshing and the breathing modes, for $\omega ={10}^4{\tilde{\omega }}_p$. Full and dashed lines: C- and L-TDDFT for amplitudes $A=0.005$, 0.5, and 0.75. At $A=0.005$, C- and L-TDDFT are identical (see Fig. 8).

Figure 12

Percent deviation of C-TDDFT from the (exact) L-TDDFT result for the time-averaged absolute value of the scaled power, ${(t_2-t_2)}^{-1}{\int }_{t_1}^{t_2}dt\mid \mathcal{P}\left(t\right)\mid ∕\omega A^2$. Top panel: sloshing mode, averaged over one cycle. Bottom panel: breathing mode, averaged over the first (full symbols) and the second half-cycle (open symbols).

Figure 13

(Color online) Snapshots of the adiabatic and nonadiabatic xc potentials during one cycle of the sloshing mode, with amplitude $A=0.75$. Black solid line: $V_{\mathrm{xc}}^{\mathrm{ALDA}}$. Red (medium gray) long-dashed, green (light gray) medium-dashed, and blue (dark gray) dotted lines: $V_{\mathrm{xc}}^{\mathrm{ALDA}}+V_{\mathrm{xc}}^M$ for frequencies $\omega =0.1{\tilde{\omega }}_p$, $\omega ={\tilde{\omega }}_p$, and $\omega =10{\tilde{\omega }}_p$.

Figure 14

(Color online) The same as Fig. 13, but for the breathing mode with amplitude $A=0.5$.

Figure 15

Comparison of ALDA (full lines) and L-TDDFT (dashed lines) xc potentials for the sloshing mode (top panel) and the breathing mode (bottom panel), both with amplitude $A=0.75$. The same snapshots during one cycle are shown as in Figs. 13, 14.