Including nonlocality in the exchange-correlation kernel from time-dependent current density functional theory: Application to the stopping power of electron liquids

We develop a scheme for building the scalar exchange-correlation (XC) kernel of time-dependent density functional theory (TDDFT) from the tensorial kernel of time-dependent current density functional theory (TDCDFT) and the Kohn-Sham current density response function. Resorting to the local approximation to the kernel of TDCDFT results in a nonlocal approximation to the kernel of TDDFT, which is free of the contradictions that plague the standard local density approximation (LDA) to TDDFT. As an application of this general scheme, we calculate the dynamical XC contribution to the stopping power of electron liquids for slow ions to find that our results are in considerably better agreement with experiment than those obtained using TDDFT in the conventional LDA.

Figure 1

(Color online) Friction coefficient of electron liquid of carbon density $(r_s=1.59)$ vs the atomic number of an ion. Green squares are results with neglect of the dynamical $\mathrm{XC}$ [Eq. (7)]. Triangles are results with the dynamical $\mathrm{XC}$ included within LDA to the conventional TDDFT [Eq. (10)]. Circles are results with the dynamical $\mathrm{XC}$ included with use of Eqs. (5, 12, 13, 14, 15).

Figure 2

(Color online) Friction coefficient of electron liquid of $r_s=2.2$ vs the atomic number of an ion. The squares are the results with neglect of the dynamical $\mathrm{XC}$ as obtained from Eq. (7). Triangles are the results with the dynamical $\mathrm{XC}$ included within LDA to the conventional TDDFT as obtained from Eq. (10). Circles are results with the dynamical $\mathrm{XC}$ included with use of Eq. (5) for $\mathrm{XC}$ kernel and Eqs. (12, 13, 14, 15) of LDA to TDCDFT. Solid squares are the measured stopping power of Al of Ref. 21 for ions $(v=0.5\text{a.u.})$ moving at a distance of $1.2\text{a.u.}$ from the last atomic plane of the Al (111) surface.

Figure 3

(Color online) Friction coefficient of electron liquid of $r_s=2$ versus the atomic number of an ion. Squares are the results with neglect of the dynamical $\mathrm{XC}$ [Eq. (7)]. Triangles are the results with the dynamical $\mathrm{XC}$ included within LDA to the conventional TDDFT [Eq. (10)]. Circles are results with the dynamical $\mathrm{XC}$ included with use of Eqs. (5, 12, 13, 14, 15). Solid squares are the measurements from Ref. 23 of stopping power of Au for ions $(v=0.68\text{a.u.})$ channeled along the (110) direction. The dotted line is the calculation of Ref. 24 with the dynamical $\mathrm{XC}$ included within the linear-response theory of the homogeneous electron gas.