Nonlocal and Nonadiabatic Effects in the Charge-Density Response of Solids: A Time-Dependent Density-Functional Approach

The charge-density response of extended materials is usually dominated by the collective oscillation of electrons, the plasmons. Beyond this feature, however, intriguing many-body effects are observed. They cannot be described by one of the most widely used approaches for the calculation of dielectric functions, which is time-dependent density functional theory (TDDFT) in the adiabatic local density approximation (ALDA). Here, we propose an approximation to the TDDFT exchange-correlation kernel which is nonadiabatic and nonlocal. It is extracted from correlated calculations in the homogeneous electron gas, where we have tabulated it for a wide range of wave vectors and frequencies. A simple mean density approximation allows one to use it in inhomogeneous materials where the density varies on a scale of 1.6 $r_s$ or faster. This kernel contains effects that are completely absent in the ALDA; in particular, it correctly describes the double plasmon in the dynamic structure factor of sodium, and it shows the characteristic low-energy peak that appears in systems with low electronic density. It also leads to an overall quantitative improvement of spectra.

Figure 1

The dynamic structure factor of sodium at $q=0.531/a_0$: experimental IXS results of Ref. [40] (red crosses) and different levels of theory for the HEG at $r_s=4$, namely ALDA (thin dashed green curve) and $2p2h$ (thin yellow solid curve) and for the real material, namely RPA (purple short-dashed curve), ALDA (green dashed curve) and $2p2h$ (yellow solid curve). Inset: zoom on the double plasmon, in particular $2p2h$ (yellow curve) with (solid curve) and without (dotted-dashed curve) Gaussian broadening. Arrow: energy of the double plasmon in Ref. [13].

Figure 2

Dynamic structure factor of the HEG for $r_s=8$ at $q=2.2k_F$. $2p2h$ results (yellow continuous curve) are compared with RPA (purple short-dashed curve), ALDA (green long-dashed curve), and with the results of Takada [15] (blue dotted-dashed curve).

Figure 3

The dynamic structure factor of silicon at $q=0.81/a_0$ compared with IXS results of Weissker et al. [12]. Curves have the same meaning as in Fig. 4, but additionally, the AMDA result is shown (blue dotted-dashed curve).

Figure 4

Similar to Fig. 1, at higher momentum transfer $q=1.161/a_0$. The homogeneous results are omitted here.