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arXiv:cond-mat/0209250

Failure of the local density approximation in time-dependent spin density functional theory

Zhixin Qian, Adi Constantinescu, Giovanni Vignale

Published Sep 10, 2002• 0 citations
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Abstract

It has been known for some time that the exchange-correlation potential in time-dependent density functional theory is an intrinsically nonlocal functional of the density as soon as one goes beyond the adiabatic approximation. In this paper we show that a much more severe nonlocality problem, with a completely different physical origin, plagues the exchange-correlation potentials in time-dependent spin density functional theory. We show how the use of the spin current density as the basic variable solves this problem, and we provide an explicit local expression for the exchange-correlation fields as functionals of the spin currents.

DOI
10.48550/arxiv.cond-mat/0209250

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It has been known for some time that the exchange-correlation potential in time-dependent density functional theory is an intrinsically nonlocal functional of the density as soon as one goes beyond the adiabatic approximation. In this paper we show that a much more severe nonlocality problem, with a completely different physical origin, plagues the exchange-correlation potentials in time-dependent spin density functional theory. We show how the use of the spin current density as the basic variable solves this problem, and we provide an explicit local expression for the exchange-correlation fields as functionals of the spin currents.
It has been known for some time that the exchange-correlation potential in time-dependent density-functional theory is an intrinsically nonlocal functional of the density as soon as one goes beyond the adiabatic approximation. In this paper we show that a much more severe nonlocality problem, with a completely different physical origin, plagues the exchange-correlation potentials in time-dependent spin-density functional theory. We show how the use of the spin current density as a basic variable solves this problem, and we provide an explicit local expression for the exchange-correlation fields as functionals of the spin currents.
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.
When a system has evolved far from a ground-state, the adiabatic approximations commonly used in time-dependent density functional theory calculations completely fail in some applications, while giving qualitatively good predictions in others, and sometimes even quantitative predictions. It is not clearly understood why this is so, and developing practical approximations going beyond the adiabatic approximation remains a challenge. This paper explores three different lines of investigation. First, an expression for the exact time-dependent exchange-correlation potential suggests that the accuracy of an adiabatic approximation is intimately related to the deviation between the natural orbital occupation numbers of the physical system and those of the Kohn-Sham system, and we explore this on some exactly-solvable model systems. The exact expression further suggests a path to go beyond the adiabatic approximations, and in the second part we discuss a newly proposed class of memory-dependent approximations developed in this way. Finally, we derive a new expression for the exact exchange-correlation potential from a coupling-constant path integration.
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).
Approximations for the ground-state exchange-correlation potential of density-functional theory have reached a high level of sophistication. By contrast, time- or frequency-dependent exchange-correlation potentials are still being treated in a local approximation. Here we propose a novel approximation scheme, which effectively brings the power of the generalized gradient approximation (GGA) and meta-GGA to time-dependent density-functional theory. The theory should allow a more accurate treatment of strongly inhomogeneous electronic systems (e.g. molecular junctions) while remaining essentially exact for slowly varying densities and slowly varying external potentials.
I review recent advances in understanding the physical character of the exchange-correlation (xc) potential in time-dependent density functional theory. I show that in an electron gas of slowly varying density the dynamical part of the xc potential is an extremely nonlocal functional of the density. This difficulty can be circumvented by introducing an xc vector potential that is a function of the local current density. The form of this vector potential is entirely specified by symmetry. Its physical effects are analogous to those of the viscous force in the Navier-Stokes equation of classical hydrodynamics. Recent applications of the xc vector potential to the calculation of infrared absorption linewidths in quantum wells are described.