Current-Dependent Exchange-Correlation Potential for Dynamical Linear Response Theory.

The frequency-dependent exchange-correlation potential, which appears in the usual Kohn-Sham formulation of a time-dependent linear response problem, is a strongly nonlocal functional of the density, so that a consistent local density approximation generally does not exist. This problem can be avoided by choosing the current-density as the basic variable in a generalized Kohn-Sham theory. This theory admits a local approximation which, for fixed frequency, is exact in the limit of slowly varying densities and perturbing potentials. 71.45Gm;73.20Dx,Mf;78.30Fs;21.10Re;36.40+d;85.42+m Typeset using REVTEX 1 Gross and Kohn (GK) [1] have applied the time dependent density functional theory (TDFT) of Runge and Gross, [2] to the case of the linear response of interacting electrons in their ground-state to a time-dependent potential v1(~r, ω)e . The objective was the determination of the linear density response, n1(~r, ω)e . They reduced the problem to a set of self-consistent single particle equations, analogous to the Kohn-Sham equations for time-independent systems [3], with an effective potential of the form v 1 (~r, ω) = v1(~r, ω) + ∫ n1(~r′, ω) |~r − ~r| d~r′ + v1xc(~r, ω); (1) the exchange-correlation (xc) potential v1xc(~r, ω) is linear in n1(~r, ω), v1xc(~r, ω) = ∫ fxc(~r, ~r ′;ω)n1(~r ′, ω)d~r′, (2) and the kernel fxc(~r, ~r ;ω) is a functional of the unperturbed ground state density n0(~r). In the spirit of the local density approximation (LDA) for static and quasi-static problems [4] , they then considered the case where both n0 and n1 are sufficiently slowly varying functions of ~r. As fxc is of short range for a homogeneous system, they proposed the following plausible approximation for systems of slowly varying n0(~r): fxc(~r, ~r ′;ω) ∼ f xc(|~r − ~r ′|, ω;n0(~r)). (3) The superscript h refers to a homogeneous electron gas and the function f xc is a property of the homogeneous electron gas [1,5]. However it was noted later by Dobson [6] that the approximation (3), when applied to an electron gas in a static harmonic potential 1 2 kr and subjected to a uniform electric field, v1(~r, ω) = −~ E · ~re , violates the so called harmonic potential theorem (HPT), related to the generalized Kohn’s theorem [7], according to which the density follows rigidly the classical motion of the center of mass: n1(~r, ω) = ~ ∇n0(~r) · ~ RCM(ω). This raised serious questions about the validity of the approximation (3). Dobson observed that one could satisfy the HPT by requiring that the GK approximation (3) be applied in a frame of reference moving with the local velocity of the electron fluid. The xc potential obtained by this construction is a functional of the current-density as well as the density [8]. 2 Further light on the problem with approximation (3) was thrown by Vignale’s observation [9] that the covariance of the time-dependent Schrödinger equation under transformation to an accelerated frame of reference requires the total force exerted on the system by the exchange-correlation and Hartree potentials to vanish, in agreement with the third law of Newtonian mechanics. This implies that the exact fxc must satisfy the sum rule [10] ∫ fxc(~r, ~r ;ω)~ ∇′n0(~r ′)d~r′ = ~ ∇v0xc(~r), (4) where v0xc(~r) is the static xc potential. This sum rule is violated by eq. (3). More generally, one can deduce [10] that fxc(~r, ~r , ω) for a non-uniform system is of long range in space and a nonlocal functional of the density distribution. These results indicate that, contrary to more optimistic expectations, a local-density approximation for time-dependent linear response in general does not exist as long as one insists on describing dynamical exchange-correlation effects in terms of the density. In this paper, we want to demonstrate however that a local approximation for the timedependent linear response theory can be constructed in terms of the current density. We consider the linear current response ~j1(~r, ω)e −iωt of interacting electrons in their groundstate to a time-dependent vector potential ~a1(~r, ω)e . This problem includes, as a special case, the scalar potential problem studied by GK, because any scalar potential v1(~r, ω) can be gauge-transformed to a longitudinal vector potential ~a1(~r, ω) = ~ ∇v1(~r, ω)/iω, and the density response can be calculated from the current response using the continuity equation n1(~r, ω) = ~ ∇·~j1(~r, ω)/iω. As usual, we express the exact induced current as the response of a non-interacting reference system (the “Kohn-Sham” system) to an effective vector potential ~a 1 = ~a1 + ~a1H + ~a1xc: j1i(~r, ω) = ∫ ∑ j χKS,ij(~r, ~r ′, ω) · ~a 1j (~r ′, ω)d~r′, (5)