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Oxford University Press eBooks

Time-dependent current-DFT

Carsten A. Ullrich

Published Dec 22, 2011• 0 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780199563029.003.0010

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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.
The formally exact expression of the exchange-correlation potential in the local-scaling density functional theory (LSDFT) was derived recently by the author, and it is equivalent to that of Holas and March. The basic equation of the LS-DFT was also derived, and it is equivalent to the differential virial theorem (DVT). The LS-DFT has been extended to the local-scaling current density functional theory (LS-CDFT). This has been compared with the formulation of Holas and March. In this note, the LS-CDFT is extended to the timedependent system; i.e., the local-scaling time-dependent current density functional theory (LS-TDCDFT) is formulated. As a straightforward extension of ref. 3, the density and the paramagnetic current density jp are adopted as the basic variables, which is the same procedure used by Colwell et al. It is explicitly shown that the time-dependent exchange-correlation scalar potential vxc and the vector one Axc are same functionals of firstand second-order density matrices as those of the time-independent case, while the total effective potentials are not so. The equation of motion of jp provides the DVT for the time-dependent system in the magnetic field. The time-dependent term of the DVT is equal to that of the nonmagnetic time-dependent system. Throughout the entire formulation, the notation of the tensor product is suppressed. Here, an N-electron system in an external scalar potential vextðr; tÞ, an external vector potential Aextðr; tÞ, and an interaction potential uðr; r0Þ 1⁄4 =jr r0j is considered, where is the coupling constant treated as a variable parameter. In a time-independent theory, the quantity to be minimized is the energy. But in the time-dependent case, the action S plays the role:
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.
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).
Time-dependent density-functional theory (TDDFT) treats dynamical exchange and correlation (xc) via a single-particle potential, Vxc(r,t){V}_{\mathrm{xc}}(\mathbf{r},t)Vxc​(r,t), defined as a nonlocal functional of the density $n({\mathbf{r}}^{\ensuremath{'}},{t}^{\ensuremath{'}})$. The popular adiabatic local-density approximation (ALDA) for Vxc(r,t){V}_{\mathrm{xc}}(\mathbf{r},t)Vxc​(r,t) uses only densities at the same space-time point (r,t)(\mathbf{r},t)(r,t). To go beyond the ALDA, two local approximations have been proposed based on quantum hydrodynamics and elasticity theory: (a) using the current as the basic variable (C-TDDFT) [G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett. 79, 4847 (1997)], (b) working in a comoving Lagrangian reference frame (L-TDDFT) [I. V. Tokatly, Phys. Rev. B 71, 165105 (2005)]. In this paper we illustrate, compare, and analyze both nonadiabatic theories for simple time-dependent model densities in the linear and nonlinear regime, for a broad range of time and frequency scales. C- and L-TDDFT are identical in certain limits, but, in general, exhibit qualitative and quantitative differences in their respective treatment of elastic and dissipative electron dynamics. In situations where the electronic density rapidly undergoes large deformations, it is found that nonadiabatic effects can become significant, causing the ALDA to break down.
A rigorous formulation of time-dependent current density functional theory (TDCDFT) on a lattice is presented. The density-to-potential mapping and the V\mathcal{V}V-representability problems are reduced to a solution of a certain nonlinear lattice Schr\"odinger equation, to which the standard existence and uniqueness results for nonlinear differential equations are applicable. For two versions of the lattice TDCDFT, we prove that any continuous-in-time current density is locally V\mathcal{V}V-representable (both interacting and noninteracting), provided that in the initial state the local kinetic energy is nonzero everywhere. In most cases of physical interest, the V\mathcal{V}V-representability should also hold globally in time. These results put the application of TDCDFT to any lattice model on a firm foundation, and pave the way for studying exact properties of exchange-correlation potentials.