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Physical Review Letters

Time-Dependent Density Functional Theory beyond Linear Response: An Exchange-Correlation Potential with Memory

John F. Dobson, M. J. Bünner, Eberhard K. U. Gross

Published Sep 8, 1997• 105 citations
Publisher

Abstract

We propose a memory form of exchange-correlation potential ${v}_{\mathrm{XC}}(\mathbf{r},t)$ for time-dependent interacting many-particle systems. Unlike previous memory-XC potentials, our ${v}_{\mathrm{XC}}$ is not limited to the linear response regime. The proposed form of ${v}_{\mathrm{XC}}$ is a generalized local-density approximation chosen so as to satisfy the nonlinear harmonic potential theorem and Newton's third law. For the case of the inhomogeneous electron gas, we give an explicit prescription for ${v}_{\mathrm{XC}}$ based solely on an existing parametrization of the linear XC response kernel ${f}_{\mathrm{XC}}^{\mathrm{hom}}(n,\ensuremath{\omega})$ of the uniform gas. Application to quantum wells seems promising.

DOI
10.1103/physrevlett.79.1905

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We propose a memory form of exchange-correlation potential vXC(r,t){v}_{\mathrm{XC}}(\mathbf{r},t)vXC​(r,t) for time-dependent interacting many-particle systems. Unlike previous memory-XC potentials, our vXC{v}_{\mathrm{XC}}vXC​ is not limited to the linear response regime. The proposed form of vXC{v}_{\mathrm{XC}}vXC​ is a generalized local-density approximation chosen so as to satisfy the nonlinear harmonic potential theorem and Newton's third law. For the case of the inhomogeneous electron gas, we give an explicit prescription for vXC{v}_{\mathrm{XC}}vXC​ based solely on an existing parametrization of the linear XC response kernel fXChom(n,\ensuremathω){f}_{\mathrm{XC}}^{\mathrm{hom}}(n,\ensuremath{\omega})fXChom​(n,\ensuremathω) of the uniform gas. Application to quantum wells seems promising.
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).
Frequency dependent exchange-correlation kernels for time-dependent density functional theory can be used to construct approximate exchange-correlation potentials. The resulting potentials are usually not translationally covariant nor do they obey the so-called zero-force condition. These two basic symmetry requirements are essential for using the potentials in actual applications (even in the linear regime). We provide two pragmatic methods for fully imposing these conditions for both linear and nonlinear regimes. As an example, we take the Gross and Kohn frequency dependent XC functional [Phys. Rev. Lett. 55, 2850 (1985)], correct it, and numerically test it on a sodium metal cluster. Violation of the basic symmetries causes instabilities or spurious low frequency modes.
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.
We have recently proposed an approximation for the dynamical exchange-correlation (XC) potentials of time-dependent current-density functional theory beyond the local density approximation [Phys. Rev. Lett. 97, 036403 (2006)]. The novel feature of the approximation is that the dependence of the dynamical XC potentials upon the density gradient and other inhomogeneity parameters (e.g., the Laplacian of the density and the kinetic energy density) is introduced by applying the generalized gradient approximation (GGA) and meta-GGA to the calculation of the XC stress tensor. The scheme may allow a more accurate treatment of the dynamical properties of an inhomogeneous system, while reducing to the exact form for slowly varying densities and slowly varying external potentials. In this work, we present in detail the derivation of this XC potential, spell out the underlying assumptions, and explain their physical basis.
Most applications of time-dependent density-functional theory (TDDFT) use the adiabatic local-density approximation (ALDA) for the dynamical exchange-correlation potential V(xc)(r,t). An exact (i.e., nonadiabatic) extension of the ground-state LDA into the dynamical regime leads to a V(xc)(r,t) with a memory, which causes the electron dynamics to become dissipative. To illustrate and explain this nonadiabatic behavior, this paper studies the dynamics of two interacting electrons on a two-dimensional quantum strip of finite size, comparing TDDFT within and beyond the ALDA with numerical solutions of the two-electron time-dependent Schrodinger equation. It is shown explicitly how dissipation arises through multiple particle-hole excitations, and how the nonadiabatic extension of the ALDA fails for finite systems but becomes correct in the thermodynamic limit.
The key quantity in the formulation of linear-response theory within time-dependent density-functional theory is the so-called exchange-correlation (xc) kernel. This chapter discusses the properties of the xc kernel. In general, it is a complex and frequency-dependent object, and a functional of the ground-state density. Several exact properties such as the zero-force theorem, static and high-frequency limit, and the long-range behaviour are discussed. The xc kernel can be related to a time-dependent variational principle provided one pays attention to certain time-dependent boundary conditions to ensure causality. The adiabatic approximation of the xc kernel is discussed. Finally, the third section of the chapter reviews the xc kernel of the homogeneous electron liquid. Many of its exact properties are covered, such as the high- and low-frequency limit and the long-wavevector limit. Three widely used parametrizations are presented and the analytical behaviour in the complex frequency plane is investigated.