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.
Time-dependent density-functional theory (TDDFT) treats dynamical exchange and correlation (xc) via a single-particle potential,
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) uses only densities at the same space-time point
(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.
We formulate equations of time-dependent density functional theory (TDDFT) in the comoving Lagrangian reference frame. The main advantage of the Lagrangian description of many-body dynamics is that in the comoving frame the current density vanishes, while the density of particles becomes independent of time. Therefore a comoving observer will see a picture which is very similar to that seen in the equilibrium system from the laboratory frame. It is shown that the most natural set of basic variables in TDDFT includes the Lagrangian coordinate, $\mathbf{\ensuremath{\xi}}$, a symmetric deformation tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$, and a skew-symmetric vorticity tensor, ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$. These three quantities, respectively, describe the translation, deformation, and the rotation of an infinitesimal fluid element. Reformulation of TDDFT in terms of new basic variables resolves the problem of nonlocality and thus allows us to regularly derive a local nonadiabatic approximation for exchange correlation (xc) potential. Stationarity of the density in the comoving frame makes the derivation to a large extent similar to the derivation of the standard static local density approximation. We present a few explicit examples of nonlinear nonadiabatic xc functionals in a form convenient for practical applications.
Today, most application of time-dependent density functional theory (TDDFT) use adiabatic exchange-correlation (XC) potentials that do not take into account non-local temporal effects. Incorporating such "memory" terms into XC potentials is complicated by the constraint that the derived force and torque densities must integrate to zero at every instance. This requirement can be met by deriving the potentials from an XC action that is Galilean in-variant (GI). We develop a class of simple but flexible forms for an action that respect these constraints. The basic idea is to formulate the action in terms of the Eularian-Lagrangian transformation (ELT) metric tensor, which is itself GI. The general form of the XC potentials in this class is then derived and the linear response limit is derived as well.
The scalar f(xc) and tensor f(xc) exchange-correlation (xc) kernels are key ingredients of the time-dependent density functional theory and the time-dependent current density functional theory, respectively. We derive a comparatively simple relation between these two kernels under the assumption that the dynamic xc can be considered "weak." A calculation of the frequency-dependent dielectric function of silicon using this relation in conjunction with Vignale-Kohn f(xc) demonstrates a potential of our method to account for the dynamic many-body effects within the rigorous scheme of time-dependent density functional theory. Our formula provides a bridge between the scalar f(xc), which directly enters many applications, and the tensor f(xc) which, due to its locality in space, is much easier to approximate.
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.