Importance of Nonlocality in the DynamicalExchange-Correlation Kernel in theTime-Dependent Density Functional Theory

in LDA to the static Kohn-Sham scheme whichamounts to the calculation without including any
dynamical xc effects.

Takada Group
The time-dependent density functional theory(TDDFT) is a powerful tool for studying excitations inatomic, molecular, and condensed-matter systems. In
the linear-response regime, the key quantity ofTDDFT is the dynamical exchange and correlation(xc) kernel fxc(r,r’; ), defined as the Fourier transformof fxc(r,r’;t-t') which is given by the functional
derivative of the Kohn-Sham xc potential Vxc(r,t) withrespect to the electron density n(r’,t’). In actual
implementation of the theory, we need to resort tosome approximation to this quantity. In the absence ofits detailed information, the adiabatic local density
approximation (ALDA) has been adopted in virtually
all calculations so far.
Some years ago, we derived a rigorous formula forobtaining the experimentally observed stopping power,–dE/dx, of solids for slow ions in the framework of
TDDFT [1], but it turns out that the formula provides
a nonzero value for the friction coefficient (or –dE/dx
divided by v the velocity of incident ions in the limitof v0) in the homogeneous electron gas with thelimit of zero electron density (or in the vacuum), if weadopt ALDA for fxc(r,r’; ). This contradictory result
urges us to develop a better approximation scheme forthis quantity beyond the level of ALDA.
In pursuit of such a scheme, we have reformulated
the problem of the stopping power in the frameworkof the time-dependent current density functional
theory (TDCDFT) [2] to find a new interesting
rigorous relation between fxc(r,r’; ) and the tensorial
xc kernel, the key quantity in TDCDFT. By invoking
the local density approximation to the tensorial xc
kernel and exploiting this relation, we haveconstructed a new approximation form for fxc(r,r’; ).
The resulting form adds a nonlocal nature to ALDA
and the nonlocality is just enough to make the result
for the friction coefficient of the electron gas freefrom the above-mentioned contradiction in thevacuum limit.
In order to check the accuracy of this new form forfxc(r,r’; ), we have recalculated the friction coefficientof the homogeneous electron gas at various electron
densities and compared the obtained results with theexperimental ones. Note that our results for the
homogeneous electron gas can be directly comparedwith the experimental ones in simple metals, in viewof the fact that at low velocities the energy loss of ionsin solids is due mainly to the stopping power ofvalence electrons.
In Fig. 1, an example of such comparisons is shownat the electron density as specified by rs=2.2 (rs: theconventional nondimensional density parameter)
corresponding to aluminum with incident ionspenetrating from its (111) surface. In the figure, wehave also plotted the results in ALDA as well as thoseFig. 1. Comparison between experiment on the friction
coefficient of aluminum for slow ions (characterized by theatomic number) obtained at the incident velocity v=0.5 atomic
units (the solid squares) and theory on the same quantity of thehomogeneous electron gas with the electron density equivalent
to the valence-electron density in aluminum in the limit of zeroion velocity (v 0). The open circles, the open triangles, andthe open squares correspond, respectively, to the results in ournew approximation scheme (LDA to CDFT), ALDA tofxc(r,r’; ), and LDA to the static Kohn-Sham scheme.

As can be seen in Fig. 1, experiment exhibits an
oscillatory behavior in the friction coefficient withincreasing the atomic number of incident ions. Thisbehavior, which is attributed to the shell structure ofvalence electrons around the ion, is reproduced well in
all the calculation schemes, but the best quantitative
agreement with experiment is obtained by our newscheme (namely, LDA to CDFT) for fxc(r,r’; ).
We expect that our new approximation formfxc(r,r’; ) awaits various applications in a wide rangeof problems concerning dynamical processes inelectronic systems.
References[1] V. U. Nazarov, J. M. Pitarke, C. S. Kim, and Y.
Takada, Phys. Rev. B 71, 121106(R):1-4 (2005).
[2] V. U. Nazarov, J. M. Pitarke, Y. Takada, G. Vignale,and Y.-C. Chang, Phys. Rev. B76, 205103:1-6 (2007).
Authors
V. U. Nazarov,a J. M. Pitarke,b Y. Takada, G. Vignalec,and Y.-C. Changa
aAcademia Sinica, Taiwan,
bEuskal Herriko Unibertsitatea, Spain
cUniversity of Missouri, USA.