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arXiv:cond-mat/9605052 · Physical Review Letters

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

Giovanni Vignale, Walter Kohn

Published Sep 2, 1996• 492 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.77.2037

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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.
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This chapter discusses time-dependent current-density-functional theory (TDCDFT). It first asks under what circumstances one should choose the current density as basic variable, instead of the particle density. There are several reasons: a current-based formulation is more suitable for constructing functionals beyond the adiabatic approximation. Furthermore, the current is the natural variable to describe coupling to electromagnetic fields and vector potentials. The basic existence theorems of TDCDFT are proved, and several exact properties and the linear-response regime are discussed. The second half of the chapter deals with the so-called Vignale-Kohn functional, which is a local and frequency-dependent approximation to the exchange-correlation vector potential. The Vignale-Kohn functional is expressed in terms of viscoelastic stresses in the electron liquid. Several applications in the linear-response regime and in real time are presented. The Vignale-Kohn functional works well for describing dissipative effects in collective, plasmon-like electron dynamics.
Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).
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The exact static and time-dependent Kohn-Sham (KS) exchange-correlation potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalized by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local-density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.