Dynamical response function in sodium and aluminum from time-dependent density-functional theory

We present a detailed study of the dynamical electronic response in bulk sodium and aluminum within time-dependent density-functional theory (TDDFT). The poor results of the random-phase approximation (RPA) and the time-dependent local-density approximation (TDLDA) in sodium are greatly improved by the approximate inclusion of the finite lifetimes of electrons and holes via a modified independent-particle polarizability, which brings the calculated spectra into good agreement with experiment. For aluminum the changes are less visible, but at some values of momentum-transfer lifetime effects are necessary to obtain qualitatively correct spectra. The double-peak structure in aluminum, induced by band-structure effects, is partially washed out by the inclusion of the finite lifetimes. The latter do not, however, create a double peak by themselves as they do in the case of the homogeneous electron gas. Studying the performance of different time-dependent and nonlocal TDDFT kernels, we conclude that the Gross-Kohn, Corradini et al., and the Hubbard local-field factors improve the spectra compared to the RPA results. However, the results agree less well with experiment than those obtained using TDLDA with added lifetime effects. These results apply to both the loss spectra and the plasmon dispersion.

Figure 1

Comparison of the experimental and the calculated dynamical structure factor for sodium. Calculations are performed in RPA and TDLDA. For reference also the HEG RPA (Lindhard) results are plotted. Note that different energy scales are used in the different panels. The plasmon cutoff vector $Q_c$ is $\sim$0.4$a_0^{-1}$.

Figure 2

Same as Fig. 1 but for aluminum. Experiments are (a) present work and (b) Schülke et al. (Ref. 13). The plasmon cutoff vector $Q_c$ is $\sim$0.6$a_0^{-1}$.

Figure 3

Comparison of experimental and calculated dynamical structure factor for sodium. In TDLDA+LT, the inclusion of lifetimes is done with the lifetimes calculated for Na, while in TDLDA+HEG-LT the lifetimes of the homogeneous electron gas have been used ($Q_c\simeq 0.4a_0^{-1}$). Note that different energy scales are used in the different panels.

Figure 4

Same as Fig. 3 but for aluminum ($Q_c\simeq 0.6a_0^{-1}$).

Figure 5

Imaginary part of the self-energy (damping, or inverse lifetime) for sodium and aluminum. The HEG curves are from Ref. 54, real-solid results for Na are those of Ref. 53, and the actual lifetimes for Al have been calculated by us with $GW$ contour deformation and averaged over the BZ.

Figure 6

Comparison of experimental and calculated dynamical structure factor for sodium. Calculation are performed with the TDLDA, the Gross-Kohn, the Hubbard, and the Corradini et al. kernels ($Q_c\simeq 0.4a_0^{-1}$). Note that different energy scales are used in the different panels.

Figure 7

Same as Fig. 6 but for aluminum ($Q_c\simeq 0.6a_0^{-1}$).

Figure 8

Plasmon dispersion of sodium (left-hand side) and aluminum (right-hand side). The plasmon cutoff vector $Q_c$ is $\sim$0.4$a_0^{-1}$ in Na and 0.6$a_0^{-1}$ in Al.